Professional Blackjack

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PROFESSIONAL BLACKJACK

STANFORD WONG Pi Vee Press

PROFESSIONAL BLACKJACK by

Stanford Wong Pi Vee Press

copyright © 1975, 1977, 1980, 1981, 1994 by Pi Yee Press

All rights reserved No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. Inquiries should be addressed to Pi Yee Press, 7910 Ivanhoe #34, La Jolla, CA 92037-4511. ISBN 0-935926-21-6 Printed in the United States of America 4 5 6 7 8 9 10

4


PREFACE

This 1994 edition of Professional Blackjack is a major update and expansion. It covers many rules variations that the previous edition did not cover. It contains simulation results to replace the "best guesses" in previous editions. Appendixes C, D, and E are new material in this edition. Parts of the previous edition were removed and expanded into another book, Black.jack Secrets. The main part involved was the chapter entitled "How to Win Without Getting Kicked Out." After you master card counting, you will want to read Blackjack Secrets to learn how to get away with it in a casino. Professional Blackjack is a reference book for card counters. It contains a virtually complete set of strategy numbers for the high-low counting system. The numbers in this book have been objectively derived on a computer. This material has been thoroughly tested in casinos throughout the world. Win rates herein are estimates based on simulations by Blackjack Count Analyzer. The win rates con-

5

tained in this book are based on simulations totaling more than ten billion hands of blackjack. You can buy this software and reproduce almost any strategy recommendation and almost any win rate in this book. Small parts of this material first appeared in one or another of the newsletters: Stanford Wong s Blackjack Newsletter, Current Blackjack News, Blackjack World, and Nevada Blackjack. (Of those, only Current Blackjack News is still published.) Thanks to the people who read pre-publication copies and whose suggestions have made this a better book: Anthony Curtis, Michael Dalton, Dave Douglas, Frank Polo, Donald Schlesinger, and John Speer. This book can be improved. If you find passages that are wrong, or if your questions go unanswered, please put your comments in writing and email them to me at [email protected] or mail them to me at Pi Vee Press, 7910 Ivanhoe #34, La Jolla, California 92037-4511. I reserve the right to publish your questions and my answers.

6


CONTENTS

CHAPTER 1 INTRODUCTION Dealer Action on Soft 17.................................................. "Pack" and "Deck" Multiple Numbers For Some Decisions Win Rates Benchmark Rules

15 16 16 17 17 18

CHAPTER 2 GENERIC BASIC STRATEGY 22 Calculating Basic Strategy Pairs Soft Hands Table 1 Generic Basic Strategy Hard Hands Surrender Insurance

CHAPTER 3 THE HIGH-LOW SYSTEM High-Low Count Table 2 High-Low Count Count Per Deck Estimating Your Edge Table 3 Player Edge

23 24 25 26 28 28 29

30 31 31 32 34 35

CONTENTS

Jumping into a Game in Progress Effect of Rules on Strategy Insurance Pairs Table 4 Split, With No Double After, S17 Table 5 Split, With Double After, S17 Doubling Down Table 6 Double Down, S17 Soft Standing Table 7 Soft Standing, S17 Hard Standing Table 8 Hard Standing, S17 If the Dealer Hits Soft Seventeen Summary Strategy Tables Importance of Strategy Tables Table 9 Strategy Numbers, S17 Table 10 Strategy Numbers, HI7

CHAPTER 4 INSURANCE Table 11 When to Take Insurance A 10-Count for Insurance Table 12 10-Count for Insurance Bustout Bet

CHAPTER 5 DOUBLE DOWN Double on Any Number of Cards Double on Two or Three Cards With Multiple-Card Surrender Table 13 Double Down, S17 Table 14 Double Down, H17 Table 15 Double, Not Last Chance, S17 Table 16 Double, Not Last Chance, HI7 Table 17 Double Down, With Multiple-Card Surrender, S 17 Table 18 Double Down, With Surrender After Doubling, S17 With Surrender After Doubling

7 36 36 37 38 38 39 40 41 42 42 43 43 44 45 45 46 48

52 53 54 54 56

58 58 59 59 60 61 62 63 64 65 66

8


CHAPTER 6 SPLIT

67

Resplits 68 Double Any Number of Cards 68 Table 19 Split, With No Double After, SI7 70 Table 20 Split, With Double After, S17 71 72 Table 21 Split, With No Double After, H17 Table 22 Split, With Double After, HI7 73 Table 23 Split, With Double Any Number of Cards 74 and No Double After Split, S17 Table 24 Split, With Double Any Number of Cards 75 and No Double After Split, HI7 Table 25 Split, With Double Any Number of Cards 76 Including After Split, S17 Table 26 Split, With Double Any Number of Cards 77 Including After Split, HI7 Table 27 Split, With Double Any Two or Three 78 Cards Before Split and Any Two After, S17 Table 28 Split, With Double Any Two or Three 79 Cards Before Split and Any Two After, HI7

CHAPTER 7 NO HOLE CARD

80

Lose Only Original Bet to a Natural 81 Lose All to a Natural 81 82 Table 29 Lose All to a Natural, S17 Any 21 Ties Dealer Natural 85 Table 30 Modifications to Appendixes When You Lose All to a Natural 85 Table 31 No Dealer Natural, S 17

CHAPTER 8 SURRENDER Early Surrender Table 32 Early Surrender Late Surrender Table 33 Late Surrender Table 34 Multiple-Card Early Surrender Multiple-Card Surrender Table 35 Multiple-Card Late Surrender Surrender After Doubling Down

86

89 90 91 92 93 94 94 95 96

CONTENTS

9

Second-Chance 21 Table 36 Second-Chance 21

96 97

CHAPTER 9 FIVE CARDS

98

Half a Bet 99 Table 37 2-Card Hands With 5-Card Half Win, 817 100 Table 38 3-Card Hands With 5-Card Half Win, 8 17 102 Table 39 4-Card Hands With 5-Card Half Win, 8 17 103 Table 40 5-Card Hands With 5-Card Half Win, 8 17 104 Table 41 2-Card Hands With 5-Card Early Half Win, 817 106 Table 42 3-Card Hands With 5-Card Early Half Win, 8 17 108 Table 43 4-Card Hands With 5-Card Early Half Win, 817 109 Table 44 5-Card Hands With 5-Card Early Half Win, 817 110 Table 45 3-Card Hands With 6-Card Automatic Win, 8 I7 111 Table 46 4-Card Hands With 6-Card Automatic Win, 817 112 Table 47 5-Card Hands With 6-Card Automatic Win, 817 11 3 Table 48 2-Card Hands With db3, 6-Card Win, HI 7 114 Table 49 3-Card Hands With db3, 6-Card Win, HI 7 116 Table 50 4-Card Hands With 6-Card Win, HI7 .. 1 17 Table 51 5-Card Hands With 6-Card Win, HI7 .. II 8 Automatic Win 1I9 Table 52 2-Card Hands With Winning 5-Card 21 Pays 2-1, 817 120

10


Table 53 3-Card Hands With Winning 5-Card 21 Pays 2-1, 817 122 Table 54 4-Card Hands With Winning 5-Card 21 Pays 2-1, S17 123 124 Double Pay on Five-Card 21

CHAPTER 10 6-7-8 ETC 6-7-8 Double Pay for 6-7-8 Table 55 Any 6-7-8 Pays Double, SI7 Table 56 6-7-8 Suited Pays Double, SI7 Triple Pay For 6-7-8 Table 57 Any 6-7-8 Pays Triple, 817 Table 58 6-7-8 Suited Pays Triple, S17 Table 59 Any 6-7-8 Pays Double, H17 Table 60 6-7-8 Suited Pays Double, H17 Table 61 Any 6-7-8 Pays Triple, HI7 Table 62 6-7-8 Suited Pays Triple, H17 7-7-7 Table 63 7-7-7 Pays Double, 817 Table 64 7-7-7 Pays Triple, S17 Table 65 7-7-7 Pays Double, HI7 Table 66 7-7-7 Pays Triple, HI7

CHAPTER 11 OVER/UNDER 13 Basic Strategy Single Deck Counting Cards Table 67 Crush Count

CHAPTER 12 HALVES COUNT Table 68 Comparison of High-Low and Halves Counts Table 69 Halves Strategy Numbers, S17 Table 70 Halves Strategy Numbers, H17

125 126 126 127 129 130 131 133 134 135 136 13 7 138 138 139 140 141

142 142 143 145 146

148 149 152 154

CHAPTER 13 DOUBLE EXPOSURE

156

Estimating Your Edge Table 71 Player Edge for Dbl Exposure

157 157

11

CONTENTS

Basic Strategy Counting Cards Table 72 Double Exposure Basic, Part 1 Table 73 Double Exposure Basic, 817 Table 74 Double Exposure Basic, H17 Table 75 High-Low, ObI Exposure, Part 1 Table 76 High-Low, ObI Exposure, S17 Table 77 High-Low, ObI Exposure, H17 Win Rates at Double Exposure Table 78 Halves, Dbl Exposure, Part 1 Table 79 Halves, Double Exposure, S17 Table 80 Halves, Double Exposure, HI7 Table 81 Double Exposure Win Rates

CHAPTER 14 FINER POINTS

158 159 160 162 164 168 170 172 175 176 178 180 182

183

Expected Win Table 82 Effects on Win Rates Risk Table 83 Area in the Tail of a Normal Distribution Table 84 Probabilities of Low Points Chance of Reaching All-Time High Avoid Going Broke: Use Proportional Betting Optimal Bet Size Table 85 Variance and Covariance for Blackjack Table 86 Optimal Bet as a Proportion of Your Advantage Overbetting Stopping Rules The Optimal Number of Simultaneous Hands All Decisions Are Approximations Which Counting System How Many Numbers To Learn Side Count of Aces Two-Card Combinations Resolving Toss-ups Quick and Accurate

183 185 189 191 197 200 201 201 203 204 206 207 208 2 11 213 215 2I7 21 7 218 2 18

12


Thorp's Complete Point Count With Other People at the Table

CHAPTER 15 SHUFFLES

219 220

222

New Decks Shuffled, But Not Together 223 Single-Deck Shuffle Study 224 Table 87 Raw Data From Ten Shuffles 226 Table 88 Card Positions of Ten Shuffles 227 Table 89 Post-Shuffle Gaps Between Initially Adj acent Cards 229 Speed of Blackjack 23 1 Table 90 Average Times for Shuffling and Dealing 232 Single-Deck Blackjack In Reno Table 91 Average Rounds Per Hour for Single-Deck Blackjack in Reno 234 Table 92 Average Times for Shuffling and Dealing Six-Deck Blackjack in Atlantic City 235 Table 93 Average Rounds Per Hour for Six-Deck Blackjack in Atlantic City 23 7 Search For Streakiness 239 Table 94 Outcomes Versus Results of Previous Two Hands 241 Table 95 Average Probabilities for Dealer's Hand, SI7 242 Table 96 Dealer's Hand Versus Two Preceding Dealer Hands 243 Table 97 One Player, $100 Bets, Six Unshuffled Decks, Lay and Pay

247

Table 98 Any Two Cards from Well-Shuftled Six249 Deck Shoes Table 99 Consecutive Cards From Unshuffled SixDeck Shoes 250 Table 100 Alternate Cards From Unshuffled SixDeck Shoes 251

APPENDIXES A & B Appendix A Table Al High-Low, One Deck, HI7

252 254 255

13

CONTENTS

Table A2 Table A3 Table A4 Appendix B Table B 1 Table B2 Table B3 Table B4

High-Low, Four Decks, H17 High-Low, One Deck, 817 High-Low, Four Decks, 817 Halves, Halves, Halves, Halves,

One Deck, H17 Four Decks, HI7 One Deck, S17 Four Decks, SI7

APPENDIX C

286

Table C 1 Frequencies and Benchmark Table C2 Frequencies and Benchmark Table C3 Frequencies and Benchmark Table C4 Frequencies and Shuffle at 62 Table C5 Frequencies and Shuffle at 42 Table C6 Frequencies and Deck Table C7 Frequencies and Exposure Benchmark Table C8 Frequencies and Exposure Halves

APPENDIX D Table Dl Frequencies 1 Deck, 817 Table D2 Frequencies Deck, S17 Table D3 Frequencies 6 Decks, S17 Table D4 Frequencies Decks, S 17

259 263 267 252 253 257 261 265

Win Rates 6-Deck 288 Win Rates 2-Deck 288 Win Rates I-Deck 288 Win Rates 6-Deck, 289 Win Rates 6-Deck, 289 Win Rates Halves, 6289 Win Rates Double 290 Win Rates Double 290

291 (per 100,000) of Initial Hands, 294

(per 100,000) of Decisions, 1 295 (per 100,000) of Initial Hands, 296 (per 100,000) of Decisions, 6 297

14


298

APPENDIX E 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 6 6

Deck, S17, Dealer Shows Ace Deck, H17, Dealer Shows Ace Deck, S17, Dealer Shows 2 Deck, H17, Dealer Shows 2 Deck, S17, Dealer Shows 3 Deck, H17, Dealer Shows 3 Deck, S17, Dealer Shows 4 Deck, HI7, Dealer Shows 4 Deck, S17, Dealer Shows 5 Deck, H17, Dealer Shows 5 Deck, S17, Dealer Shows 6 Deck, HI7, Dealer Shows 6 Deck, Dealer Shows 7 Deck, Dealer Shows 8 Deck, Dealer Shows 9 Deck, Dealer Shows 10 Decks, S17, Dealer Shows Ace Decks, HI7, Dealer Shows Ace Decks, S17, Dealer Shows 2 Decks, H17, Dealer Shows 2 Decks, S17, Dealer Shows 3 Decks, H17, Dealer Shows 3 Decks, S17, Dealer Shows 4 Decks, H17, Dealer Shows 4 Decks, S17, Dealer Shows 5 Decks, H17, Dealer Shows 5 Decks, S17, Dealer Shows 6

302 303 304 305 306 307 308 309 310 311 312 313 3 14 3 15 3 16 3 17 3 18 3 19 320 32 1 322 323 324 325 326 327 328

6 Decks, HI?, Dealer Shows 6

329

6 6 6 6

33 0 33 1 332 333

Decks, Decks, Decks, Decks,

Dealer Dealer Dealer Dealer

Shows Shows Shows Shows

7 8 9 10

GLOSSARy SELECTED REFERENCES

334

342

INDEX 346 PUBLICATIONS BY STANFORD WONG. 349

15

CHAPTER 1 INTRODUCTION

The purpose of this book is to serve as a handy reference to card counters. When you travel, if there is any chance you will encounter a blackjack game, take this book with you. Whatever rules you encounter at blackjack, you should be able to open this book and find the strategy numbers you need to attack the game. Chapter 2 presents generic basic strategy. Chapter 3 presents the high-low counting system for one commonly-found set of rules. Following that are chapters presenting strategy indexes for insurance, double down, splitting, the no-hole-card game, surrender, multiplecard bonuses, and bonuses for particular hands such as 6-7-8 of the same suit. You ought to be able to find the strategy index numbers you need for the rules you face in the casino of your choice.

16


Chapter 11 is devoted to over/under 13, a recent innovation. Over/under 13 is two side bets. You can bet that your first two cards will total over thirteen or under thirteen. Aces count as one and not eleven, and thirteen loses. Over/under 13 can be profitably exploited by card counters. Chapter 12 presents a more powerful counting system, the halves. Chapter 13 is devoted to double exposure. This is a form of blackjack in which both the dealer's cards are exposed before the players act, and ties go to the dealer. Chapter 14 explains some of the finer points of blackjack, including expected win, risk, optimal betting, and counting in a casino. Chapter 15 presents several studies of casino shuffles. The appendix contains tables of strategy indexes, expectations, and frequencies of particular hands.

Dealer Action on Soft 17 Some of the tables in this book have been derived for dealer stands on soft seventeen, and some have been derived for dealer hits soft seventeen. An abbreviation is used to distinguish between the two: s17 means dealer stands on soft seventeen, and hI7 means dealer hits soft seventeen.

"Pack" and "Deck" This book uses "deck" to mean 52 cards, and "pack" to mean a collection of cards that might or might not contain exactly 52 cards. The reason for this usage is to

17

INTRODUCTION

make the advice on card counting unambiguous: "count per deck" means "count per 52 cards."

Multiple Numbers For Some Decisions Some decisions have more than one number associated with them. For example, sometimes with 4-4 it is correct to hit at very low counts, double down at very high counts, and split at intermediate counts. Where this complication exists, adequate explanation is provided in the text. Another example: With soft eighteen you may want to stand, you may want to hit, or you may want to double down. This complication generally is handled by having two lines in the table, one for stand versus double and one for hit versus stand.

Win Rates This book contains numerous win rates. They are based on computer simulations of at least 30 million hands of blackjack each. Their purpose is to aid in understanding the importance of various rules options. Benchmark Rules I have selected a benchmark, an arbitrary set of rules, playing conditions, and bets. This benchmark is shown on the next page. Use of a different benchmark would result in different numbers. However, the relative importance of different rules variations would be approximately the same no matter what set of rules were used as the benchmark. The benchmark betting scheme is not the best way to bet. Much better is to leave the table on negative

18


Benchmark

Rules

Six decks. One-deck cut, meaning five decks are dealt out. Dealer stands on soft seventeen. Double down is allowed on any first two cards but not after splitting. Resplits are allowed to a total of four hands. Split aces receive one card each, and cannot be resplit. Insurance is offered, but no other side bets. No surrender. Naturals pay 3:2, and tie a natural by the dealer. The high-low counting system is used, with strategy numbers from -10 to +10. There is a total of two players at the table. The bet scheme is $100 at counts per deck of +4 or more, $75 at +3, $50 at +2, $25 at zero or +1, and $10 at all negative counts.

Benchmark Win rate: $16 Std dev: $415

INTRODUCTION

19

counts. On positive counts, you must bet in a manner that does not attract unwanted attention to yourself. See Blackjack Secrets for in-depth betting advice. Simulating the base set of rules and betting scheme finds a win rate that is a benchmark for evaluating the cost or benefit of various deviations from the base rules. Benchmark Win Rate The benchmark win rate is $16 per 100 hands. In a casino you will be able to play 50 to 300 hands per hour, so 100 hands represents roughly an hour's play. Throughout this book results are stated in dollars per hour, which really means dollars per 100 hands. Sampling Error That $16 win rate is a simulation result, and thus is an estimate. Though it is an estimate for 600 million hands of blackjack, it still contains what statisticians call "sampling error." That is, a different 600-millionhand simulation would result in almost but not quite the same number. The term used to describe the precision of an estimate is the standard error. The standard error applicable to the $16 win rate is $0.20. The actual simulation result was $15.78, which was rounded because the pennies and dimes are not significant. So you should think of the $16 as meaning somewhere in the $15.38 to $16.18 range. It would be nice to have a smaller standard error of course, and the way to do that is to increase the sample size. That is the reason why the sample size was run up to 600 million hands - to reduce the standard error to a reasonable number. Each of the win rates in this book is based on a sample of at least 30 million hands, and has a standard error of less than $1. So if one rule is changed from the benchmark, and a simulation with the new rule yields

20


a win rate of say $20 per hour, that rule increased the win rate by $4 per hour. That $4 can be thought of as the value per 100 hands to a card counter of that rule change. But it is not exactly $4 per 100 hands because both the $20 and the $16 are sample statistics with sampling error. If you run your own simulation trying to duplicate these results you probably would get $4, but you could possibly get $3 or $5 as the value of the rule. And of course if you use a different bet scheme, a different counting system, make an occasional error, or play against a different number of decks or with a different number of players at the table, or with different penetration, the value to you might be different from $4 per hour. Average Bet Size The average bet size for the benchmark is $26.54. That is the average initial bet; it does not include additional amounts wagered on doubles, splits, and side bets such as insurance. That $26.54, plus or minus a penny or two, is the average bet size on all simulations reported in this book for six decks, penetration down to the last deck, and the high-low system.

Benchmark Risk The standard deviation for 100 hands for the benchmark is $415. That is, the win rate may be $16 but any

given 100 hands typically has a result considerably less than or more than $16. In repeated samples of 100 hands, 2/3 of the samples have wins of $16 plus or minus $415. Risk varies with the rules. Some rule changes mean more risk, and some mean less risk. Being able to double down more often (such as after splitting pairs) means more risk, as represented by a higher standard devia-

INTRODUCTION

21

tion. Restrictions on doubling down mean a lower win rate but that is partially offset by lower risk. Win Rates Reported A small box is used to indicate a win rate, and is the result of a simulation of at least 30 million hands. The box explains the manner in which the rules of this simulation differ from the benchmark, the win rate per 100 hands using the benchmark betting scheme, and the standard deviation applicable to 100 hands. The rest of the information pertaining to that simulation is the same as the benchmark as discussed above. As an example, if one deck with penetration 26 cards is used instead of six decks, but everything else kept the same as the benchmark, the win rate is $48 per 100 hands. A summary of win rates is presented in table 82 (page 185). Double-exposure win rates are presented separately, as table 81 (page 182).

One deck Win rate: $48 Std dev: $482

22


CHAPTER 2 GENERIC BASIC STRATEGY

This chapter contains generic basic strategy, and is a copy of chapter 2 of Basic Blackjack and Blackjack Secrets. If you already know basic strategy, skip this chapter. Basic strategy is the best way to play a blackjack hand on the first round after a shuflle, assuming you see

no cards other than your own and the dealer's upcard. For a person who does not count cards, basic strategy is the best way to play every hand. This chapter presents basic strategy for single exposure, which is blackjack where the dealer has one card face up for you to see as you are playing your hand. Basic strategy is what plays you should make if you are not counting cards and you do not have any information about the dealer's hole card. It is presumed that you

GENERIC BASIC STRATEGY

23

know the total in your own hand and the dealer's upcard, but no other cards. Chapter 13 contains basic strategy for double exposure, which is blackjack where the dealer has two cards face up. Another book, Basic Blackjack, presents basic strategy for all the various rules that have been offered for blackjack, such as dealers take ties on seventeen, six-card hand pays double, surrender after doubling down, etc.

Calculating Basic Strategy Basic strategy can be either total-dependent or composition-dependent. Total-dependent means the strategy numbers require only the dealer's card and the total points in your hand. Composition-dependent means the strategy numbers require knowledge of the dealer's card and the precise cards that make up your hand. For example, total-dependent strategy says stand on twelve against 4. Composition-dependent strategy for twelve against 4 requires you to specify how you get to twelve: Do you have 7-5, 8-4, 3-2-2-5, or what? If you get to twelve by 10-2 or 2-10 (where 10 means any 10-count card), and two or fewer decks are being used (or seven or fewer if the dealer stands on soft seventeen), you should hit. If you get to twelve by any other route, or enough decks are being used, you should stand. See Peter Griffin's The Theory of Blackjack for a good discussion of compositiondependent strategy. (The 10-2 versus 4 advice is from page 176 of Griffm's book.) There are few differences between compositiondependent and total-dependent strategies for single deck, and none that are important for multiple decks.

24


(The more decks shuffled together, the less difference one card makes.) This book uses total-dependent strategy. The details of basic strategy depend on the particulars of the rules. However, you must start someplace. This chapter presents a version of basic strategy that is approximately correct for the most common sets of rules - a generic basic strategy. For modifications of this generic basic strategy that are appropriate for different sets of rules, see Basic Blackjack. Blackjack is most commonly played with the dealer's hand showing one card face up. If you are playing blackjack in a game where you get to see two cards face up in front of the dealer before you play your hand, go to chapter 13 for playing-strategy advice. Table 1 presents generic basic strategy. It contains advice for every decision the blackjack player commonly makes. Each column is a different dealer upcard. (10, J, Q, and K are lumped together as 10.) Each row is a different player hand. Technically, table 1 is basic strategy for multiple decks and dealer stands on soft seventeen.

Pairs The order of decisions presented in table 1 is the order in which you evaluate your hand. Do you have a pair? At most casinos, any two 10count cards, e.g. J-K, are a pair and may be split. If you have a pair, the first part of table 1 tells you how to play your hand. Use this part of the table to decide whether to split your pair. To split means to make another bet equal in size to your fIrst bet, and play each card as the start of a separate hand.


25

If you split a pair and catch another card of the same value, resplit if you can. If it is correct to split a pair, it is correct to resplit. You mayor may not be allowed to double down after splitting a pair. For example, if you split 8-8 and catch a 3 for eleven, you mayor may not be allowed to double down on that eleven. If doubling down after splitting is allowed, then splitting is more attractive and you should split more often. The first part of table 1 assumes you are not allowed to double down after splitting. If you are allowed to double down after splitting, then use the last part of table 1 - the part on the facing page. The decisions that are hits in the first part of table 1 and splits on the facing page are: 6-6 against 2, 4-4 against 5 or 6, 3-3 against 2 or 3, and 2-2 against 2 or 3.

Soft Hands Do you have an ace? Aces count your choice of either eleven or one. A hand in which an ace counts eleven is called a soft hand, and the total points in it are called a soft total. The second part of table 1 explains how to play soft hands. The double-down advice is broken down into db and dbs. The reason is you need to know what to do with a particular total if you cannot double down. For example, suppose you have soft eighteen and the dealer shows 3. Your best play is to double down, so that is what you do if you can. But if your soft eighteen is a three-card hand, say ace-2-5, then you probably will not be allowed to double down. Table 1 lists "dbs" for that hand, which means if you are not allowed to double down then you should stand.

26


Table 1 Generic Basic Strategy Player's Dealer's Upcard 7 Hand 2 3 4 5 6 8 double not allowed after split A-A spl spl spl spl spI spl spl

10-10 9-9 8-8 7-7 6-6 5-5 4-4 3-3 2-2 soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 soft 14 soft 13 hard 21 hard 20 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12 11 10 9 8

7 6 5

spl spl spl h db h

spl spl spl spl db h

h

h h

h

h h h h h

spl spl spl spl db

spl spl spl spl db h h spl spl spl spl

spl spl spl spl db h

spl spl

dbs dbs dbs dbs db db db db h db db db h db db db h db db h h h h db

spl spl h db h spl spl

h db db db

h

h h h

h

h

h

h

h db h

h h

spl

10

A

spl spl

spl spl spl spl h h h h h h db h h h h h h h h h h h

h

h h h

h

h

h

h h

h

h

h

h

h h

h h

h h

h h

h

h h h

h h h h h db db h

sr

sr sr

sr

h

h

h db db h h

spl spl

9

db db db

db db db

db db db

h db db h

h h h h

h h h h

h h h h

h h h h

h

h h

h h

h h h

h h h

db db

db h

h h h h h

h h h h h

h

h

h h h

h

h h h

h h h h h h h

27


Table 1 Continued Dealer's Upcard Player's 6 8 5 7 3 Hand 2 4 double allowed after split A-A spl spl spl spl spl spl spl 10-10 9-9 8-8 7-7 6-6 5-5

4-4 3-3 2-2

spl spl spl spl db

spl spl spl spl db

h

h

spl spl

spl spl

spl spl spl spl db

spl spl spl spl db h spl spl spl spl spl

spl spl spl spl db spl spl spl

9

10

A

spl

spl spl

spl spl

spl spl

spl spl

h

h h

h h

db

db

db

h

h h h

h h h

spl spl

spl spl

KEY

db dbs

Stand. Double down; if you cannot double, then hit. Double down; if you cannot double, then stand.

h

Hit.

spl

Split.

sr

Surrender; if you cannot surrender, then hit.

h h h h

h h h h

h h

h

h

28


Note that sometimes it is correct to hit eighteen. If you have soft eighteen and the dealer shows 9, 10, or ace, then hitting your soft eighteen is better than standing on it. One decision that is very close is soft thirteen against 5. It does not matter whether you hit or double down.

Hard Hands The lower two parts of table 1 explain how to play the rest of your hands~ Hands labeled "hard" might contain aces, but under the circumstances all such aces are counted as one. Hands tabulated as from 5 to 11 do not contain an ace; if a hand totaling eleven or less has an ace it is a soft hand and is played according to the "soft" part of the table.

Surrender Surrender means losing half a bet for the privilege of not playing out the hand. Late surrender means you cannot surrender if the dealer has a natural. The table of generic basic strategy includes strategy for late surrender. Most of the' value of late surrender comes from surrendering sixteen against 10. Late surrender is worth 0.1 % to a basic-strategy player. If you are playing blackjack at a casino that does not offer surrender, or if you are not allowed to surrender due to having more than two cards, then hit those hands for which table 1 advises surrender.


29

Insurance Table 1 does not show insurance. Basic strategy says never take insurance. Even if you have a natural, you are better off not insuring it. You are better off winning 3:2 most of the time than winning even money for sure.

30


CHAPTER 3 THE HIGH-LOW SYSTEM

How many dealer upcards can you see before you play your hand? If you see two, go to chapter 12, which discusses double exposure. This chapter discusses singleexposure blackjack. Winning at blackjack requires two things: You must

bet more when you have the advantage and less when the dealer has the advantage; and you must make correct decisions on insurance, surrender, splitting pairs, doubling down, and hitting or standing. Parts of this chapter are similar to chapter 3 of Blackjack Secrets. One difference is that this book gives "surrender" its own chapter. Another difference is that this book presents high-low strategy numbers for a

31

HIGH-LOW SYSTEM

wider range of counts per deck, -10 to +10, instead of -1 to +6.

High-Low Count You need a counting system to tell whether you have the advantage and to aid in making decisions. Aces and lOs favor you because naturals are worth half again more to you than they are to the dealer. Small cards favor the dealer by decreasing the dealer's chance of busting. The high-low system, first introduced in 1963 by Harvey Dubner, is both simple and powerful. Edward O. Thorp in the revised edition of Beat the Dealer, Lawrence Revere in Playing Blackjack as a Business, and Julian Braun in How to Play Winning Blackjack, present strategy numbers for it. This chapter contains my independent calculations for it. Peter Griffin in The Theory Of Blackjack says the betting correlation of the high-low system is 0.97 and the playing correlation is 0.51. Counting cards in the high-low system is relatively simple. Start with a count of zero after the cards are shufiled. Add one for every small card (2, 3, 4, 5, 6) that

Table 2 High-Low Count Card

Count

2,3,4,5,6 7,8,9 1O,J,Q,K,Ace

+1 0 -1

32


you see. Subtract one for every ace or 10-count card that you see. Do not change your count for 7, 8, or 9. This is shown in table 2. Keep a running count. Accumulate the total since the last shufile. A full deck contains the same number of +1 cards as -1 cards. Therefore, at the end of the deck the running count should come back to zero. This is called a "balanced count."

Count Per Deck This section explains count per deck, also called true count. The high-low count tells you when the best cards, the 1Os and aces, outnumber the small cards. The richer the pack, that is, the more 1Os and aces relative to small cards, the better for you. You need to know how rich the pack is in order to make appropriate bets and decisions in play. Richness of the pack depends on the proportion of excess 1Os and aces. For decision purposes, you must relate the running count to the number of decks you have not seen. For example, twelve lOs and aces remaining to be used when the dealer is halfway through a single deck is two 1Os and aces more than average; that is as favorable to the player as if there were four excess 1Os and aces in 52 cards, and is described as a count per deck of +4. For betting and playing decisions, a running count of + 1 with half a deck remaining is equivalent to a running count of +2 with one deck remaining, to a running count of +4 with two decks remaining, and to a running count of +8 with four decks remaining. Therefore, you must convert the running count into count per

33

HIGH-LOW SYSTEM

deck for making decisions. Simply divide the running count by the number of decks (or fraction of a deck) that you have not seen. If less than one deck remains, your count per deck will be greater than your running count. If more than one deck remains, your count per deck will be less than your running count. For example, suppose you approach a blackjack table with a single-deck game in progress and see the following cards.

First Player Second Player 2-3-2-4-7

10-10

Dealer 3-8-9

You glance at the cards and see that the running count is +3. Since what you have not seen is almost one deck, the count per deck is slightly above +3. The dealer picks up the cards, and you place a bet. You receive 2-10, the dealer's upcard is 2, the first other player has a natural, and you do not see the second other player's cards. You must decide whether to hit or to stand. The running count is now +2. Since about 2/3 of the deck remains unseen, you divide by 2/3. This is the same as multiplying by 3/2. It gives a count per deck of +3. So you stand instead of hitting. You need only approximate the count per deck. You do not need an exact count of the number of cards remaining. If you had seen roughly half a deck in a single-deck game, a running count of +2 would translate to a count per deck of about +4. If you had seen about half a deck (26 cards) in a double-deck game, a running count of +2 would translate to a count per deck slightly greater than +1 (2 divided by 1.5 gives 4/3). A rough estimate of the count per deck suffices for decisions because you

34


need to know only whether it exceeds an integer in a table.

Estimating Your Edge Your advantage or disadvantage if you play basic strategy varies with the rules and number of decks used. Current Blackjack News is a good source for keeping up to date with the rules at various casinos and your advantage or disadvantage. Commonly you are at a disadvantage of about 0.5% with basic strategy. Table 3 contains more detail on estimating the casino's edge right after a shuffle. The starting point depends on the number of decks used. Simulation shows that a basic-strategy player playing with the benchmark rules has an expectation of 0.00% for one deck, -0.32% for two decks, and -0.5% for four or more decks. The negative signs mean the casino has an edge. The rest of table 3 contains numbers that you can add to one of those starting points, depending on how the rules you face vary from those benchmark rules. Early surrender gives the player 0.7%, of which 0.4% is against an ace and 0.3% is against 10. Table 3 allows you to find your expectation for the first hand after a shuffle. These are simulation results, and thus contain a small amount of random error. To a small extent, the effect of a given rule also depends on the number of decks used. The best presentation of how your edge after a shuftle depends on the rules is in Peter Griffin's The Theory of Blackjack. You also can generate such numbers yourself with Blackjack Count Analyzer if you ask for a shuffle after every hand. Table 3 applies to the first hand after a shuflle. For subsequent hands, your edge might be higher or lower.

HIGH-LOW SYSTEM

35

Table 3 Player Edge benchmark: benchmark: benchmark: benchmark: benchmark:

rule

one deck two decks fOUf decks six decks eight decks

0.00% -0.32 -0.47 -0.52 -0.55

to the above add the following: change in player edge

dealer hits soft seventeen double after splits double ten & eleven only double nine, ten & eleven only resplit aces (if four or more decks) late surrender early surrender lose all on doubles/splits versus natural

-0.20% O. 14 -0.17 -0.08 0.08 0.08 0.71 -0.11

For each point increase in the count per deck, your advantage goes up by 0.5%. (It goes up by increasing amounts at high counts, presumably due to insurance becoming more valuable the higher the count.) Generally, in a multiple-deck game, when you have a count per deck of +1 you are playing even with the casino - no advantage or disadvantage. At a count per deck of +3 you have an advantage of about 1%. The 0.5% per count per deck works both ways: The dealer's advantage over you increases with negative counts. At a count per deck of -1 you are at a disadvantage of about 1% - if you play perfectly. If the count per

36


deck is -4, you can expect to lose at a rate of about 2.5%, which is 2% faster than if the dealer were to shuftle. Winning with card counting requires betting more money when you have an edge than when you are at a disadvantage, and playing your hands correctly.

Jumping into a Game in Progress When the best available table is a game in progress and you want to jump in without waiting for a shuflle, just start counting from zero and play according to your count. Treat the unseen discards the same as cards yet to be dealt. An unseen card is an unseen card whether it has already been used or remains to be dealt. Example: You jump into a four-deck shoe after about one deck has been used. You play and count as another deck is used. If your running count is +6, what is your count per deck? Two of the four decks have been used, but you have counted the cards in only one deck. Since about three decks remain unseen by you, divide your running count by three to get a count per deck of +2.

Effect of Ru les on Strategy Some casinos use one deck, some use six, and some use other multiples. Strategy numbers vary only slightly with the number of decks used. The tables in this chapter should be thought of as based on multiple decks. They actually are based on four decks, but are almost exactly the same as strategy numbers based on other multiples such as two, six, etc. One-deck strategy numbers are slightly different from multiple-deck strategy numbers. You can use Blackjack Count Analyzer soft-

HIGH-LOW SYSTEM

37

ware to find strategy numbers tailored to any number of decks and any set of rules. You should truncate, and not round, when using these tables. Examples: If the count per deck required to double down is +4, then double down only if you have a count per deck of +4 or more, and do not double down if your count per deck is say 3.8. If a count per deck of -1 is required to stand, then stand if your count per deck, truncated, is -lor higher. (Zero is higher than -1, and a count per deck of -1.8 is truncated to -1.) At some casinos dealers stand on soft seventeen, whereas at other casinos dealers hit it. The discussion of this chapter initially assumes that the dealer stands on soft seventeen, and then also covers what to do when the dealer hits soft seventeen. The decisions are discussed in the order in which you make them at a casino: insurance, pair splits, double down, and hit or stand. The strategy numbers in this book are derived for multiple decks, and are also very close for one deck. You must memorize them; when you are playing blackjack in a casino, you can refer to these strategy numbers only in your head.

Insurance Advice of well-meaning but ill-informed gamblers that you should insure only a natural is worth its cost - nothing; you should buy insurance if more than onethird of the unseen cards are lOs. As soon as you see the dealer's ace, begin considering whether you should buy insurance. If the count per deck is greater than or equal to +3, buy insurance.

38


Pairs Table 4 covers pair splits, assuming that you cannot double down after splitting. If you can double down after splitting, then use table 5 instead. Count per deck also determines whether you should split pairs. The higher the count per deck, the more pairs you should split. The exception is 8-8 against 10, which you should split when the count per deck is less than +6 and not split when the count per deck is +6 or more.

Table 4 Split, With No Double After, S17 Player's Hand ace-ace 10-10 9-9

8-8 7-7 6-6

2

spl

3 spl

8 0 spl -9 2

5-5 4-4 3-3

8

2-2

7

-2

spl spl 0 3 3

Dealer's Upcard 5 6 7 spl spl spl -9 5 4 6 -4 -4 -3 6 spl spl spl spl spl spl spl spl -5 -7 -3 4

0 0

-2 -4

-9 -9

8

9

10

-8

-7

-8

A -3

-8

-9 spl

6*

3 spl

spl

spl spl

KEY blank spl s17 number

*

Do not split. Split. Dealer stands on soft seventeen. Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number. Reverse the meaning. Split only if the count per deck is less than the number in the table.

39

HIGH-LOW SYSTEM

If you do not split, then play your hand according to the strategy number for the total. Examples: A) You get 3-3 against 3. The number from table 4 is +3. If the count per deck equals or exceeds +3, you should split the pair. If the count per deck is less than +3, you should hit. B) You get 3-3 against 9. That spot in table 4 is blank, so do not split, no matter what the count per deck. C) You get 8-8 against 9; table 4 says to split no matter what the count per deck.

Table 5 Split, With Double After, S17 Player's Hand ace-ace 10-10

2 spl

3 spl

9-9 8-8

-2 spl

7-7

-9

-3 spl spl

6-6

-1

-4

8

Dealer's Upcard 5 7 6 spl spl -9 5 4 6 -5 3 -6 -6 spl spl spl spl spl spl spl spl -6 -8 -10 4 spl

8

9

10

A

-8

-7

-8

-3

8*

spl

-8

-9

spl

spl

3

5

5-5

4-4 3-3 2-2

0 -3

6 -4 -5

1 -7 -7

-1

-4

-9 -9

spl spl

spl spl

4 5

KEY

blank spl

Do not split. Split.

s17

Dealer stands on soft seventeen.

number Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number.

*

Reverse the meaning. Split only if the count per deck is less than the number in the table.

40


Doubling Down Doubling down also depends on count per deck. You should be more likely to double down 'if the count per deck is higher. If you decide to double down, always put up additional money equal to the full amount of your original bet. The soft-doubling parts of all double-down tables in this book assume that the ace in your hand can be valued at either 1 or 11 after you split. If the ace cannot swing, then do not double down on soft hands of nineteen or less. Examples: A) You get 6-5 against 4. Your hand totals eleven, and table 6 says to double down no matter what the count per deck. B) You get 4-5 against 9. Your hand totals nine and table 6 says do not double down, so you take a hit. e) You get ace-8 against 6. You may call your total nine and double down if you wish, and table 6 advises doing so if the count per deck equals or exceeds +1. If the count per deck is less than +1, or if doubling and catching a 2 would give you eleven instead of twenty-one, call your total nineteen and stand.

41

HIGH-LOW SYSTEM

Table 6 Double Down, S17 Player's Hand

2

3

11 10 9 8** 7

db -8

db -9

1

0

ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2

10

KEY blank db s17

9

8 0 1

8 5 -2 -3 4 7 7 7

Dealer's Upcard 7 5 6 db db db -9 -10 db db -6 -6 3 -2 -4 1 5 3 9 9 4

6

5

4

3 -6 -7

1

1 -10

-2 0 1

3

-8 -10

-6 -4 -1 0

8 -6

9 -4

-4

-1

10 -4 4

A 1

4

7

db db -9 -4 -1

Do not double down. Double down. Dealer stands on soft seventeen.

number Double down if the count per deck equals or exceeds the number; do not double down if the count per deck is less than the number. **

With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

42


Soft Standing Table 7 summarizes soft standing decisions. Soft doubling as in table 6 takes precedence, so use table 7 only if you are not doubling down. If you do not double down you should hit a soft total of seventeen or less and stand with a soft total of nineteen or more. With soft eighteen if you do not double down you should stand against 2 through 8, and hit against 9 or 10. Against ace you should hit soft eighteen when the count per deck is less than +1, but stand when the count per deck is +1 or more.

Table 7 Soft Standing, S17 Player's Hand soft 19 soft 18 soft 17 KEY blank h s 17

2

3

4

h

h

h

Dealer's Upcard 5 6 7 8 h

h

h

h

9

10

A

h h

h h

1 h

Stand. Hit. Dealer stands on soft seventeen.

number Stand if the count per deck exceeds the number; hit if the count per deck is less than the number; flip a coin if the count per deck equals the number.

43

HIGH-LOW SYSTEM

Hard Standing Table 8 covers hard standing. With a total of eleven or less, never stand. With a hard total of twelve or more, you should be more likely to stand when the count per deck is higher. Examples: A) You have 9-8 for a total of seventeen against 2; table 8 says to stand. B) You have 10-4 for fourteen against 8; table 8 says to hit. e) You have 2-6 for a total of eight against 10; you ask for a hit. Suppose you receive a 7 to bring your total to fifteen. If the count per deck is less than +4, you should ask for another hit; if the count per deck equals or exceeds +4, you should stand.

Table 8 Hard Standing, S17 Player's Hand

hard hard hard hard hard hard

2

3

4

Dealer's Upcard 5 6 7 8

9

10

17

16 15 14 13 12

KEY blank h s17 number

A

-6 -8

-10

-5

-6

-3

-4

0

-1 2

3

-7 -6 -3 0

-9 -7 -4 -1

-9 -7 -4 0

9

7

10

10

h h h

h h h

5 8 h h h

0

8

4

10

h h h

h h h

Stand. Hit. Dealer stands on soft seventeen. Stand if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

44


If the Dealer Hits Soft Seventeen Tables 4-8 assume that the dealer stands on soft seventeen. If you are playing blackjack at a casino where the dealer hits soft seventeen, you should play slightly differently, primarily against an ace. The differences against an ace are: You will be more likely to hit soft eighteen, more likely to stand on hard sixteen or hard fifteen, more likely to double down on eleven, and more likely to split 9-9. Hitting soft seventeen has these effects on strategy because the dealer is less likely to finish the hand with exactly seventeen and is more likely to bust. The major changes to make to tables 4-8 if the dealer hits soft seventeen are as follows: Table 7: Always hit soft 18 against ace. Table 8: With hard 16 against ace, stand if the count per deck is +3 or more. With hard 15 against ace, stand if the count per deck is +5 or more. With hard 12 against 6, stand if the count per deck is -3 or more. Simulation of the dealer hitting soft seventeen results in a win mte lower than the benchmark by $5 per 100 hands. So the dealer hitting soft seventeen is a rule

variation that is costly to the card counter.

Benchmark Win rate: $16 Std dev: $415

Dlr hits soft 17 Win rate: $11 Std dev: $416

45

HIGH-LOW SYSTEM

Summary Strategy Tables Tables 9 and 10 summarize strategy numbers for the high-low count. Table 9 applies if the dealer stands on soft seventeen, and table 10 applies if the dealer hits soft seventeen. Table 10 includes the major changes listed above plus minor changes. Soft eighteen is represented by two lines; the -/db line is for deciding whether to stand or double down, and the h/- line is for choosing between hit and stand. The line for doubling takes precedence. Note that in table 9, the 1 for ace-7 against ace is for choosing between hit and stand, and does not say to double down.

Importance of Strategy Tables This chapter has presented multiple-deck playing strategy, but at some casinos blackjack is dealt with one deck. Multiple-deck strategy numbers are only slightly different from single-deck strategy numbers. You will win almost as much by using multiple-deck strategy numbers with one deck as you will win by using onedeck strategy numbers with one deck. With freshly shuftled cards, the casino has a larger advantage as the number of decks increases. You might be wondering how important bet variation is relative to strategy variation. Using basic strategy and flat $26.54 bets, sitting through shoe after shoe, loses at the rate of $15 per hour. So the total improvement from using the high-low system is from -$15 to +$16, or $31. Playing hands according to' basic strategy and using the high-low for bet variation wins at the rate of $12 per hour. So bet variation alone improves on flat-betting by (text continues on page 50)

46


Table 9 Strategy Numbers, S17 Player's Dealer's Upcard 7 9 Hand 2 3 4 5 6 8 double not allowed after split spl spl spl spl spl -9 -8 -7 A-A 10-10 8 6 5 4 0 -2 -8 9-9 -4 -4 6 -9 -3 8-8 spl spl spl spl spl spl spl spl -9 spl spl spl spl spl h h 7-7 0 6-6 2 -3 -5 h h h -7 never split - play as total of ten 5-5 never split - play as total of eight 4-4 0 -2 -9 spl 8 3 h h 3-3 h 7 3 0 -4 -9 spl h 2-2 6 ace-9 10 8 5 4 3 ace-8 8 5 1 1 -2 0 h ace-7 -/db -6 -8 -10 ace-7 hIh ace-6 -7 -10 db 1 h h h -3 4 -2 -6 db ace-5 h h h h 0 -4 ace-4 h 7 -9 h h h ace-3 -1 h 7 1 -4 h h h ace-2 3 h 7 -1 h h h 0 hard 20 hard 19 hard 18 hard 17 -8 -10 hard 16 9 7 5 hard hard hard hard

11 10 9

8** 7

6 5

15 14 13 12

-6 -4

-5 -3 0 3

-1 2

db -8 1 h h h h

db -9 0 9 h h h

-7 -6 -3 0

-9

-9

-7

-7

-4 -1

-4 0

10 h h h

db -10

db db

db db

-9

-6

-4

-6

-4

-2 5

-4

-6

3 9 h h

1 9 h h

3 h h h h

h h h h

-1 h h h h h

h h h

10 h h h

8 h h h

7

10

A

-8

-3

6*

3 spl

h h

h h

h h

h h

h h h h h h h

h/1 h h h h h

-6 0 h h h

8 10 h h h

-4 4

4

h h h h h

h h h h h

4

1

47

HIGH-LOW SYSTEM

Table 9 Continued Dealer's Upcard Player's 9 5 6 7 8 Hand 3 4 2 double allowed after split -7 -8 spl spl spl spl spl -9 A-A 6 5 8 4 10-10 -6 -9 9-9 -3 -5 -6 3 -8 -2 8-8 spl spl spl spl spl spl spl spl 7-7 5 h -9 spl spl spl spl spl -8 -10 h h 6-6 -1 -4 -6 h never split - play as total of ten 5-5 h 1 -1 -4 h 6 h 4-4** h -9 spl spl h 3-3 0 -4 -7 4 2-2 -7 -9 spl spl h -3 -5 5

10

A

-8

-3

8* h h

3 spl h h

h h h

h h h

KEY

Stand. Do not hit. See the -/db row for the number saying whether to stand or double down. db Double down. Hit. h h/Do not double. See the h/- row for the number saying whether to hit or to stand. spl Split. s17 Dealer stands on soft seventeen. number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. (Note that the 1 for ace-7 against ace is hit/stand, and does not say to double down.) Reverse the meaning. Split only if the count per deck * is less than the number in the table. With 4-4, splitting is better than doubling down if ** doubling is allowed after splitting.

-/db

48


Table 10

Strategy Numbers, HI7 Player's Hand double not A-A 10-10 9-9 8-8 7-7

Dealer's Upcard 6 7 8 5 4 ai/owed after split

9

10

A

-7

-8

-4

6* h h

2 -1 h h

h h

h h

h h h h h h h

h h h h h h h

2

3

spl

spl

spl

spl

spl

6 -3

5 -4

4

-1

8 -2

-6

6

-8

-9

spl

spl

spl spl

spl spl

spl spl

spl

spl

6-6

-8 1

spl spl

5-5 4-4 3-3 2-2

7

-10 -7 -3 -5 h -1 never split - play as total never split - play as total -5 -2 3 spl 0 -4 3 -7 spl 0

8

ace-9 ace-8 ace-7 -/db ace-7 hIace-6 ace-5 ace-4 ace-3 ace-2 hard hard hard hard hard hard hard hard hard ] ]

10 9 8** 7 6 5

20 19 18 17 16 15 14 13 12

10 8 0

8 5 -2

1

-3 3 6 6

h h h h

7

6 3 -6

5 1 -8

4 0 db

-7 -3 0 1 3

-10 -6 -4 -1 0

db db -10 -5 -2

-9

-8

h h h h of ten of eight h h h h

h h h h h

h h h h h

h h h h h h h

9 10 h h h

7 10 h h h

5 8 h h h

0 4 h h h

-9 -6 3 h h h h

-6

-4 -1 h h h h h

-4 4 h h h h h

-4 -9 -5 -3 0 3

-10 -7 -5 -2 1

-6 -3 0

-9 -7 -4 -1

db -8 1

db -9 0 9 h h h

db -10 -2 5 h h h

db db -4 3 9 h h

h h h h

-8

-9 -7 -3 db db -6 1 9 h h

-4 7 h h h h

3 5 9 h h 0 3 h h h h h

49

HIGH-LOW SYSTEM

Table 10 Continued Dealer's Upcard Player's 8 5 6 7 3 2 4 Hand double allowed after split -8 spl spl spl spl spl -9 A-A 4 5 8 6 10-10 -5 -6 -8 3 -8 9-9 -2 -4 8-8

7-7

spl -10

6-6 5-5 4-4** 3-3

-2

2-2

-2

h 0

spl spl -4

spl spl -6

spl spl -8

spl spl -10

spl spl h

never split - play as total -6 1 -1 h 6 -3 -7 -9 spl spl -5 -7 -9 spl spl

spl 5 h

9

10

A

-7

-8

-4

8* h h

1 -1 h h

h h h

h h h

-9 spl h h

of ten h 4 5

h h h

KEY Stand. db

Double down.

h

Hit.

h17

Dealer hits soft seventeen.

h/-

Do not double. See the hI- row for the number saying whether to hit or to stand.

spl

Split.

number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number.

* **

Reverse the meaning. Split only if the count per deck is less than the number in the table. With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

50


$27 per hour, or 87% of the $31 gain from both bet variation and strategy variation. Playing only positive counts and leaving the table on negative counts, as popularized by earlier editions of Professional Blackjack, is a form of bet variation. Using flat $26.54 bets hand after hand but varying strategy according to the count loses at the rate of $14.34 per hour. That number comes from the same 600million-hand simulation that gave the benchmark win rate; the standard error on it is $0.14. In the box below, the -$14.34 is rounded off to -$14. So varying strategy with the count per deck improves on basic strategy by only a small amount when flat bets are used in a six-deck game. Most of that improvement is due to taking insurance, standing on hard sixteen against 10, and standing on hard fifteen against 10, all at the appropriate times of course. After those three decisions, the amounts a card counter gains by correctly deviating from basic strategy drop off dramatically. See Donald Schlesinger's "Attacking the Shoe" in the September 1986 issue of Blackjack Forum for more detail on this point.

Flat $26.54 bets, basic strategy Win rate: -$15 Std dev: $301

Flat $26.54 bets, vary strategy with count Win rate: -$14 Std dev: $300

Vary bets with count, basic strategy Win rate: $12 Std dev: $401

HIGH-LOW SYSTEM

51

As a practical matter, you do not really need a wide range of strategy numbers. I simulated the benchmark modified to use only strategy numbers -1 to +6, and got a win rate of $1 7 per hour. Using a smaller range of strategy numbers should result in a lower win rate, but I got a higher rate. I increased the sample sizes to see whether the gap would disappear or the difference would become statistically significant. It became significant. This is a puzzle, and I am continuing to work on it. My suggestion for beginners, and for anyone else who wants to cut down on the volume of numbers to memorize, is ignore all strategy numbers smaller than -1 and larger than +6. For winning money at multiple-deck blackjack, bet variation is much more important than strategy variation. In fact, all of these tables on deviating from basic strategy are of minor value. More important is to avoid being barred and to find good rules and good penetration. Appendix A contains more details on strategy rules for the high-low system.

52


CHAPTER 4 INSURANCE

Insurance is a side bet that the dealer has a natural. Table 11 gives the minimum counts per deck for which insurance is a good buy, as a function of number of decks shuffled together. For example, if six decks are used, insurance is likely to be profitable if the count per deck, after counting the dealer's ace and as many other cards as you can see, exceeds 3.0; that 3.0 applies to the sixdeck game no matter how many decks are yet unseen. My suggestion to simplify the strategy numbers:

Use 1.4 for one deck, 2.4 for two decks, and 3 for more than two decks.

No insurance Win rate: $14 Std dev: $416

53

INSURANCE

Table 11 VVhen to Take Insurance Decks High-Low

Halves

1

1.4

1.7

2 3

2.4 2.7

2.7 3.0

4

2.9

3.2

5 6

3.0 3.0

3.3 3.3

7 8 infinite

3.1 3.1 3.3

3.4 3.4 3.7

Peter Griffin and Arnold Snyder assisted with the mathematics involved in calculations for table 11. Table 11 also includes advice for the halves counting system, which is explained in chapter 12. Take insurance at a count per deck equal to or greater than the strategy number shown in table 11. The insurance number does not depend on whether or not the dealer takes a hole card. For six decks, ignoring insurance wins about $2 per 100 hands less than the benchmark. For single-deck games, insurance is quite valuable. The following boxes report the results of two simulations, one with and one without insurance. The shuflle point is 26 cards yet unseen.

One deck, ins. Win rate: $48 Std dev: $482

One deck, no ins. Win rate: $41 Std dev: $484

54


A 10-Count for Insurance It is possible to use a IO-count to make better insurance decisions. The IO-count is the best count for insurance. The only way to make better insurance decisions is to get a glimpse of the dealer's hole card. Using a IO-count for insurance in addition to the highlow for betting and playing the hands adds about $2 per hour to the hourly win rate for six decks. At single deck, using a IO-count in addition to the high-low adds about $8 per hour to the hourly win rate. The simplest way to do the 10-count is to keep the following as a second count as shown in table 12-: assign the value +1 to all cards except 10 and assign the value -2 to every 10. If this running IO-count gets above four times the number of decks shuftled together (+4 with a single deck, +8 with a double deck, +16 with four decks, or +24 with six decks), buying insurance is profitable. Note that the 10-count insurance decision is based on the running count and not on the count per deck. For example, to buy insurance on a six-deck game you need a IO-count of +24 or more no matter how many decks have yet to be dealt. Also note that the pluses and minuses do not balance out - when all the cards are gone, your running count will be four times the number of decks used.

Table 12 to-Count for Insurance Card ace, 2, 3, 4, 5, 6, 7, 8, 9 IO,J,Q,K

Count +1 -2

INSURANCE

55

Suppose the dealer shuflles a single deck and deals the following cards: ace-9 6-6-8 7-10 After that round the high-low running count is zero. Suppose you place a bet, are dealt 8-9, and the dealer shows an ace. Would you buy insurance? Though the high-low count of -1 says no, you should. The reason is that an excess of 7s, 8s, 9s, and aces has appeared. Of the 42 cards you have not seen, 15 are lOs; when you buy insurance you will be paid 2 to 1 for something that will happen 15 out of 42 times, giving you an expected-win rate of roughly 7%. If you are keeping a IO-count for insurance decisions, the IO-count would be +7 and thus indicate that you should buy insurance. A single running count that relates 1Os to non-lOs for accuracy on insurance could be used for all decisions, but such a count is less accurate for estimating advantage; gains resulting from accuracy on insurance are more than offset by losses on accuracy on estimating advantage. A simulation with a IO-count for all decisions shows that it falls short of the benchmark by $8 per hour. For single deck, use of a IO-count for all decisions yields results short of the single-deck benchmark by $19 per hour! IO-count strategies are obsolete for this reason.

6 decks, add 10-count for insurance Win rate: $18 Std dev: $415

1 deck, add lO-count for insurance Win rate: $56 Std dev: $483

56


6 decks, 10-count only Win rate: $7 Std dev: $336

1 deck, 10-count only Win rate: $29 Std dev: $412

Do not use table 11 if you are keeping a lO-count, because the lO-count gives positively accurate insurance guidance.

Bustout Bet Some casinos offer a bustout bet. Winning bustout bets pay 2:1. If the dealer's first two cards total twelve through sixteen and you still have a hand in play, you can bet that the next card will be a 10 or face card. If you bust, or if you have a natural for which you have already been paid, you do not have the option of making the bustout bet. If you split or double down, the amount you can bet on the bustout is not doubled. If you use the high-low or halves (see chapter 12) counting system with the bustout bet, use 3.33 as the decision number. The reason the insurance number does not apply to the bustout bet is the dealer does not show an ace. With basic strategy, you will have the option of making the bustout bet 32% of the time, or four times as often as the dealer shows an ace. The max on the bustout bet is your original bet. So the bustout bet is approximately eight times as profitable as insurance. With insurance the dealer always shows an ace, and insurance is a better bet the more non-lOs have been removed

57

INSURANCE

from the pack. But with the bustout bet, you get to see more cards before deciding whether to make the bet. My suggestion is to play bust-out blackjack with a partner, keeping two counts and helping each other make decisions. One of you uses a ten-count to make correct bust-out and insurance bets, and the other uses a conventional counting system (such as high-low) for betting and making decisions other than bustout and insurance.

Bustout, six decks Win rate: $30 Std dey: $517

Bustout, one deck Win rate: $95 Std dey: $606

One deck, add 10-count for bustout & insurance Win rate: $107 Std dey: $583

Six decks, add 10-count for bustout & insurance Win rate: $41 Std dey: $496

58


CHAPTER 5 DOUBLE DOWN

Doubling down when the dealer stands on soft seventeen was presented as table 6 (page 41), and is reproduced here for your convenience as table 13. Table 14 is doubling down when the dealer hits soft seventeen. As you can see by comparing those two tables, how the dealer plays soft seventeen has almost no effect on whether you should double down. If you can double down on certain totals only, then use only the appropriate part of the tables. For example, if you can double down on ten or eleven only, then ignore the parts of the tables that give advice on doubling down on nine or less.

Double on Any Number of Cards Tables 13 and 14 are for use when this is your only chance to double, and if you hit instead you no longer

DOUBLE DOWN

59

have the option of doubling. This is the most common form of doubling down. At a few casinos, you can double down on any number of cards. This rule makes doubling on small twocard hands less attractive. The strategy numbers for doubling down when you can double on any number of cards are shown in tables 15 and 16. With ace-2 or ace3 when the count is high enough, the tables say to double down against 4, 5, or 6; however, you might as well just hit all hands of A-2 and A-3 because the value of doubling is approximately equal to the value of hitting no matter how high the count. That is true no matter which action the dealer takes on soft seventeen.

Double on Two or Three Cards At Las Vegas Club in Las Vegas, starting in 1993, you can double down on two or three cards, except after splitting. This requires two double-down strategies. One strategy (table 16) is for use if you will still have the option of doubling down after hitting once, and the other strategy (table 14) is for use if this is your last chance to double on this hand. For example, with eight against 5 and a high count, you ought to hit if you will have the option of doubling after that hit, but double down now if you will not have the option of doubling after hitting.

With Multiple-Card Surrender In some Asian casinos you can (or could) surrender on any number of cards. This makes hitting more attractive relative to doubling down. The appropriate doubling strategy is shown in table 17. The only change from table 13 is the strategy number for ten against 10 is +6. (text continues on page 66)

60


Table 13 Double Down, S17 Player's Hand

2

3

11 10

db -8

db -9

1

0

9

8**

9

7

ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2 KEY blank db s17 number

**

10 8 0 1

8 5 -2

-3 4 7 7 7

Dealer's Upcard 7 5 6 db db -9 -6 -10 db db 3 -4 -6 -2 3 1 5 9 9 4 db

6 3 -6 -7 -2 0 1

3

5 1

-8 -10

-6 -4 -1 0

8

-6 -4 7

9 -4 -1

10 -4 4

A 1

4

4 1 -10

db db -9 -4 -1

Do not double down. Double down. Dealer stands on soft seventeen. Double down if the count per deck equals or exceeds the number; do not double down if the count per deck is less than the number. With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

Double 10 & 11 Win rate: $10 Std dey: $400

Double 9, 10 & 11 Win rate: $12 Std dey: $406

61

DOUBLE DOWN

Table 14 Double Down, HI7 Player's Hand

2

11 10 9 8** 7

db -8

ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2

10

KEY blank db h17 number

**

1

8 0 1

3 db -9 0 9

Dealer's Upcard 7 5 6 -9 db db db -10 db db -6 -4 3 -2 -6 3 1 5 9 8 4

8 5

6

-2

-6

-3 3 6

-7 -3

3

6

0 1

7

3

5 1 -8 -10

8 -6 -4 7

9 -4 -1

10 -4 4

A 0

3

4 0

-6

db db db

-4

-10

-1 0

-5 -2

Do not double down. Double down. Dealer hits soft seventeen. Double down if the count per deck equals or exceeds the number; do not double down if the count per deck is less than the number. With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

62


Table 15 Double, Not Last Chance, Player's Hand 11 10

9 8** 7 soft soft soft soft soft soft soft soft soft

21 20 19 18 17 16 15 14 13

db

3 db

-8 2

-9 0

2

10 8 0 1

8 5 -2 -3

4 db -10 -1

6 3 -6 -7 0

8 8 9


db db -2 9

db db -4

5 1 -8 -10 -5 -1 3 4

4 1 -10 db db -6 0 7

-9 -6 4

-6 -4 8

S17

9

10

A

-4 -1

-4 4

1 4

8

KEY

blank

Do not double down.

db

Double down.

s17


number Double down if the count per deck equals or exceeds the number; do not double down if the count per deck is less than the number.

**

Double down on 4-4 rather than split only if doubling down is not permitted after pair splitting.

63

DOUBLE DOWN

Table 16 Double, Not Last Chance, Ht7 Player's Hand 11 10 9 8** 7

soft soft soft soft soft soft soft soft soft

21 20 19 18 17 16 15 14 13

Dealer's Upcard 2

3

4

5

6

7

8

9

db -8 2

db -9 0

db -10 -1

db db -2 9

db db -4 8

-9 -6 4

-6 -4 8

-4 -1

10 8 0 1

8 5 -2 -3

6 3 -6 -7 0 5 8 9

5 1 -8 -10 -5 -1 2 4

4 0 db db db -7 -1 2

10 -4 4

A 0 3

KEY blank

Do not double down.

db

Double down.

h17


number Double down if the count per deck equals or exceeds the number; do not double down if the count per deck is less than the number. **

Double down on 4-4 rather than split only if doubling down is not permitted after pair splitting.

64


Table 17 Double Down, With Multiple-Card Surrender, S17 Player's Hand 11 10 9 8** 7

ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2 KEY blank

2 db -8

3 db -9

1

0

10

8 0 1

9

4 db -10 -2 5

8 5 -2 -3 4 7 7 7

6 3 -6 -7 -2 0 1 3

Dealer's Upcard 7 5 6 db db -9 db db -6 -4 -6 3 3 1 9 9

5 1 -8 -10 -6 -4 -1 0

8 -6 -4 7

9 -4 -1

10 -4 6

A 1 4

4 1 -10 db db -9 -4 -1

Do not double down.

db Double down. s!7 Dealer stands on soft seventeen. number Double down if the count per deck equals or exceeds the number; do not double down if the count per deck is less than the number. With 4-4, splitting is better than doubling down if ** doubling is allowed after splitting.

65

DOUBLE DOWN

Table 18 Double Down, With Surrender After Doubling, S17 Player's Hand 11 10 9

2 db -8

3 db -9

1

0

8** 7 ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2

9

10

8 0 1

8 5 -2 -3 4 7 7 7

Dealer's Upcard 7 6 5 -9 db db db -6 -10 db db -6 -2 -4 3 1 5 3 9 9 4

6 3 -6 -7 -2 0 1 3

5 1 -8 -10

-6 -4 -1 0

8 -7 -4 6

9

-6 -2

10 -5

A -5

0

-1

4 1 -10

db db -9 -4 -1

KEY

blank db s17 number

Do not double down. Double down. Dealer stands on soft seventeen. Double down if the count per deck equals or exceeds the number; do not double down if the count per deck is less than the number.

**


66


For dealers hit soft seventeen, use the ace column of table 14 with the rest of table 17.

With Surrender After Doubling If you have multiple-card surrender plus surrender after doubling down, use the strategy numbers from table 18. Surrender strategy can be found in the surrender chapter. If the dealer hits instead of stands on soft seventeen, the only change to the ace column of table 18 is the strategy number for doubling down on ten against ace; it should be o.

67

CHAPTER 6 SPLIT

Table 4 (page 38) covers pair splits, assuming that the dealer stands on soft seventeen and you cannot double down after splitting; it is reproduced here as table 19. Table 5 (page 39) covers pair splits if the dealer stands on soft seventeen and you can double down after splitting; it is reproduced here as table 20. Tables 21 and 22 are parallel, and apply if the dealer hits soft seventeen. Table 21 contains the strategy numbers for splits if the dealer hits soft seventeen and you cannot double down after splitting. Table 22 contains the strategy numbers for splits if the dealer hits soft seventeen and you can double down after splitting. Note that the asterisk reverses the meaning of the strategy number. Normally you split if the count per deck equals or exceeds the number. But if there is an

68


asterisk, you split if the count per deck is less than the number. Comparison of the four tables with each other shows that being able to double down on any two cards after splitting has large effects on playing strategy, but whether the dealer hits or stands on soft seventeen makes a difference only for splitting 8-8 against ace. Double after splits is a valuable option, adding $4 per hour to the benchmark win rate.

Resplits Resplits make no difference in the sense that the strategy for splits is the same whether or not resplits are allowed. Resplits follow split strategy: The strategy number for resplits is the same as the strategy number for splits. Not being able to resplit costs you about a dollar per 100 hands.

Double Any Number of Cards Suppose you can double down on any number of cards. This rule makes splitting small pairs such as 3-3 less attractive, because you might hit the total (6) with a small card, such as a 4 or 5, bringing your hand to a total that would be an attractive three-card double. Tables 23 and 24 contain strategy numbers for splitting if you can double down on any number of cards, but cannot double down after splitting. Table 23 is for dealer stands on soft seventeen; and table 24 is for dealer hits soft seventeen. Being able to double down on any number of cards increases the value of hitting small totals, including

SPLIT

69

pairs. But being able to double down on any number of cards after splitting increases the value of splitting. The stronger effect is the increased value of being able to double down after a hit. Therefore if you can double down on any number of cards, you are less likely to split small pairs. Tables 25 and 26 contain strategy numbers for splitting if you can double down on any number of cards after splitting; table 25 is for dealer stands on soft seventeen; and table 26 is for dealer hits soft seventeen. In 1993 Las Vegas Club in Las Vegas introduced a hybrid: you can double down on any two or three cards before you split, but you can double down only on any two cards after you split. You cannot double down on four or more cards, and after splitting you cannot double down on three cards. Tables 27 and 28 cover Las Vegas Club splitting. Table 27 is for dealer stands on soft seventeen, and table 28 is for dealer hits soft seventeen.

(text continues on page 80)

No resplits Win rate: $15 Std dev: $413

70


Table 19 Split, With No Double After, S17 Player's Hand ace-ace 10-10

9-9 8-8 7-7 6-6 5-5

2 spl

3

spl

8 0 spl

-9

-2 spl spl

2

0

8 7

3 3

Dealer's Upcard 5 6 7 spl spl -9 5 4 -3 6 -4 -4 spl spl spl spl spl spl spl spl -3 -5 -7 4 spl 6

8 -8

9 -7

-8

-9

spl

spl

10 -8

A -3

6*

spl

3

4-4 3-3 2-2

0

-2

0

-4

-9 -9

spl spl

KEY blank

Do not split.

spl

Split.

s17


number Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number. *


71

SPLIT

Table 20 Split, With Double After, S17 Player's Hand ace-ace 10-10

9-9 8-8

2 spl

spl

-2

8 -3

spl

7-7 6-6 5-5 4-4

-9

3-3

0

2-2

-3

-1

3

spl spl -4

Dealer's Upcard 7 5 6 spl spl -9 4 5 -6 -6 3 -5 spl spl spl spl spl spl spl spl

4 spl 6

-6

-8

-10

6

1

-1

-4 -5

-7 -7

-9 -9

-4 spl spl

spl spl

8

-8

9 -7

-8

-9

spl 5

spl

10

A

-8

-3

8*

spl

3

4 5

KEY blank spl s 17 number

*

Do not split. Split. Dealer stands on soft seventeen. Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number. Reverse the meaning. Split only if the count per deck is less than the number in the table.

Double after split Win rate: $20 Std dev: $421

72


Table 21 Split, With No Double After, HI7 Player's Hand ace-ace

3 spl 8

1

-10 -1

4 spl 6 -3 spl spl -3

8 7

3 3

0 0

2 spl

10-10 9-9

-1

-2

8-8 7-7 6-6 5-5

spl -8

spl

Dealer's Upcard 7 5 6 spl spl -9 4 5 6 -4 -6 spl spl spl spl spl spl -5 -7

8 -8

9 -7

-8 spl

spl

10 -8

-4

6*

-1

A

2

-9

4-4 3-3 2-2

KEY blank h17 spl number

*

-2

-4

-5 -7

spl spl

Do not split. Dealer hits soft seventeen. Split. Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number. Reverse the meaning. Split only if the count per deck is less than the number in the table.

73

SPLIT

Table 22 Split, With Double After, Ht7 Player's Hand ace-ace 10-10

2 spl

3 spl

-2

8 -4

spl -10

spl spl

-2

-4

3-3

0

2-2

-2

6 -3 -5

9-9 8-8 7-7 6-6 5-5

4-4

Dealer's Upcard 7 6 8 5 spl spl spl -9 -8 4 6 5 -6 3 -5 -8 -8 spl spl spl spl spl spl spl spl spl 5 -6 -8 -10 4

1 -7 -7

-1

-9 -9

-6 spl spl

spl spl

9 -7

10

A

-8

-4

8*

1 -1

-9 spl

4 5

KEY blank

Do not split.

h17


spl

Split.


*


74


Table 23 Split, With Double Any Number of Cards and No Double After Split, S17 Player's Hand ace-ace

2 spl

10-10

9-9 8-8

0 spl

7-7

-9

6-6 5-5 4-4 3-3

2

3 spl 8 -2 spl spl 0

9 8

2-2

Dealer's Upcard 5 6 7 spl spl spl -9 4 6 5 -3 -4 -4 6 spl spl spl spl spl spl spl spl -3 -5 -7 4

6 4

2 1

2 0

8

9

-8

-7

-8 spl

spl

10 -8

A -3

6*

3 spl

-9

2 5

KEY

blank

Do not split.

spl

Split.

s17


number Split if the count per deck number; do not split if the the number. Reverse the meaning. Split * is less than the number in

equals or exceeds the count per deck is less than only if the count per deck the table.

75

SPLIT

Table 24 Split, With Double Any Number of Cards and No Double After Split, HI7 Player's Hand ace-ace 10-10 9-9

2

3

spl

spl

-1

-2

8-8

spl

spl

7-7 6-6 5-5 4-4 3-3 2-2

-8

-10

I

-I

8

Dealer's Upcard 7 5 6 spl spl -9 6 5 4 -3 -4 -6 6 spl spl spl spl spl spl spl spl -3 -5 -7

4 spl

9

5

2

8

4

I

0 0

8

-8 -8 spl

9 -7

-9 spl

10

-8

A -4

6*

2 -1

2

5

KEY blank hl7 spl number

*


76


Table 25 Split, With Double Any Number of Cards Including After Split, S17 Player's Hand ace-ace 10-10

2 spl

3 spl

9-9 8-8

-2 spl

7-7 6-6 5-5 4-4 3-3 2-2

-9 -1

-3 spl spl -4

4 spl 6 -5 spl spl -6

3 0

0 -3

-3 -6

8

9

Dealer's Upcard 5 6 7 spl spl -9 5 4 -6 -6 3 spl spl spl spl spl spl -8 -10

3 -5

-8

6 spl spl

8

-8

9 -7

-8

-9

spl 4

spl

-8

A -3

8*

3 spl

10

spl spl

KEY

blank spl

Do not split. Split.

s17 Dealer stands on soft seventeen. number Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number. * Reverse the meaning. Split only if the count per deck is less than the number in the table.

Double any number of cards, including after splitting Win rate: $26 Std dey: $424

77

SPLIT

Table 26 Split, With Double Any Number of Cards Including After Split, HI7 Player's Hand ace-ace 10-10 9-9

8-8 7-7

6-6

2 spl

-2 spl -10 -2

3 spl

8 -4

4 spl 6

Dealer's Upcard 8 5 6 7 spl spl -9 -8

5 -6

4 -8

spl spl

-5 spl spl

spl spl

-4

-6

-8

spl spl -10

8 0

-3

3 -5

-8

-3

-6

-8

-10

9

10

A

-7

-8

-4

8*

1 -1

3

-8

-9

spl spl

spl

spl

4

5-5

4-4 3-3 2-2

3 0

0

spl spl

KEY blank

Do not split.

h17


spl

Split.


*


78


Table 27 Split, With Double Any Two or Three Cards Before Split and Any Two After, S17 Player's Hand ace-ace 10-10

2 spl

9-9 8-8

-2 spl

7-7 6-6

-9

spl spl

-1

-3

3-3

3

2-2

2

0 -1

3

spl

8 -3

Dealer's Upcard 5 6 7 spl spl -9 5 4 -6 -6 3 -5 spl spl spl spl spl spl spl spl -7 -10 -5 4 spl 6

8

-8

9 -7

-8

-9

spl 4

spl

10

A

-8

-3

8*

spl

3

5-5 4-4

KEY blank

9 -2

-5

3 -4 -7

6 spl spl

spl spl

Do not split.

spl

Split.

s17



*


Double three cards before splitting, two after Win rate: $28 Std dev: $425

79

SPLIT

Table 28 Split, With Double Any Two or Three Cards Before Split and Any Two After, H17 Player's Hand ace-ace 10-10 9-9 8-8 7-7

6-6

2 spl

3 spl 8

-4

-2 spl -10 -1

spl spl -3

3 2

0 -1

Dealer's Upcard 7 5 6 spl spl -9 5 4 6 3 -5 -8 -6 spl spl spl spl spl spl spl spl -6 -7 -10

4 spl

8

9

-8

-7

10 -8

A -4

-8 spl

-9 spl

8*

1 -1

4

5-5

4-4 3-3 2-2

8 -2

3

0

-4

-6

-4

-7

-10

spl spl

KEY

blank h17 spl number

*


80


CHAPTER 7 NO HOLE CARD

At some casinos the dealer takes only an upcard, and no hole card. The players get two cards each to start, and finish their hands in the normal mamier. Then the dealer takes a second card, and if necessary, more cards. Suppose the dealer gets a natural, a two-card twentyone. At most casinos you lose unless you also have a natural; if you have gotten to twenty-one in three or more cards you lose if the dealer gets a natural. At some casinos you lose only your original bet and not any extra amounts you have wagered on doubles or splits. At other casinos if you have doubled down or split and the dealer gets a natural you lose your whole doubled bet. There also are casinos that allow you to keep your money (i.e. not lose) if you hit to twenty-one in any

No

HOLE CARD

81

number of cards and then the dealer's ace or 10 turns into a two-card twenty-one.

Lose Only Original Bet to a Natural Whether the dealer's second card is face down under the upcard or is in the pack yet to be dealt should not matter to you. This is true even if you are the last player to act. If you not taking a card means it will be the dealer's second card, the proper play of your hand is the same as if the dealer already had a second card. If you have sixteen and the proper play is to hit, then you should hit whether the dealer has a hole card or not. What does make a difference in strategy is what happens when the dealer gets a natural.

Lose All to a Natural In Europe and a few other places, the dealer does not take a hole card, and if the dealer ends up with a natural you lose your whole bet if you have doubled down or split a pair. This means you should be less willing to wager extra money on your hand when the dealer shows a 10 or ace. If you cannot lose more than your original bet in the event of a natural by the dealer, this section does not

apply. This section applies only for the situation where if you put out an extra bet and the dealer has a natural, you lose your original bet and the extra bet too. The dealer winning all on naturals does not affect your hit/stand decisions. If it is correct to hit your hand when the dealer takes a hole card, it is also correct to hit your hand if the dealer does not take a hole card. The fact

82


Table 29 Lose All to a Natural, 817 Dealer's Upcard Player's Hand 5 6 7 8 2 3 4 double not a/lowed after split A-A spl spl spl spl spl -9 -8 5 10-10 6 8 4 6 9-9 0 -8 -2 -3 -4 -4

8-8

spl

7-7 6-6 5-5 4-4

-9

3-3 2-2

8 7

ace-9 ace-8 ace-7 -/db ace-7 hIace-6 ace-5 ace-4 ace-3 ace-2 hard 20 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12 11 10 9

10 8

8** 7 6

5

2

spl spl

spl spl -3

spl spl

-5 0 never split never split 3 0 -2 0 -4 3

8

6

3

0

5 -2

1 h h h h

-3 4 7 7 7

-8 -5 -3 0 3 db

-10

-6 -7

-2 0 1

3

5 1 -8 -10 -6 -4 -1 0

spl spl

spl spl

spl

9

10

A

-7

-6

h

h/h h

h/h h

h h

h h

h h

h/-

-9

spl

h h -7 h h h play as total of ten play as total of eight -9 spl h h -9 spl h h 4 1 -10 h h db h h h db h h h -9 h h h -4 h h h -1 h h h

h h h h h

1 h h h h h

-6

-8 1 h h h h

-6

-7

-4 -1

-6

2

0 db -10 -2 5 h

db -9 0 9 h h

h

-3

h h

-9 -7 -4 -1

-9 -7 -4 0

db db -4 3 9 h h

db db -6 1 9 h h

9 10 h h h -9 -6 3 h h h h

7 10 h h h -6 -4 7 h h h h

5 8 h h h -4 -1 h h h h h

0 4

h h h

3 h h h h h h

8 10 h h h h h h h h h h

No

83

HOLE CARD

Table 29 Continued Dealer's Upcard Player's 9 7 8 5 6 3 Hand 2 4 double allowed after split spl spl spl spl spl -9 A-A -8 -7 4 5 6 8 10-10 -9 3 -3 -5 9-9 -2 -6 -6 -8 spl spl spl spl spl spl spl spl 8-8 5 h -9 spl spl spl spl spl 7-7 h h -1 -4 6-6 -6 -8 -10 h never split - play as total of ten 5-5 h -4 h 6 1 -1 h 4-4** h h 3-3 0 -4 -7 -9 spl spl 4 h 2-2 -3 -5 -7 -9 spl spl 5

10

A

-6

h

h/h h

h/h h

h h h

h h h

KEY

db h h/spl s17 number

**

Stand. Double down. Hit. See elsewhere in the table for hit or stand. Split. Dealer stands on soft seventeen. Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. (Note that the 1 for ace-7 against ace is hit/stand, and does not say to double down.) With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

84


that the next card out of the shoe could go to either you or the dealer does not affect the play of your hand. The dealer winning all on naturals also does not affect the play of your hand if the dealer shows other than a 10 or ace. The tables in this chapter have colwnns for dealer showing 2-9, but those columns are identical to those in table 9 (page 46). Basic strategy, if you lose all to a natural by the dealer, is to split ace-ace against 10 but otherwise do not double down or split against 10 or ace. Advice for card counters if the dealer might have a natural to beat your doubled bet is contained in table 29. Note that the only doubling down to consider against a 10 or ace is with eleven against 10. The only splitting to consider against a 10 or ace is ace-ace against 10. The strategy numbers for playing against 10 and ace do not depend on whether you can double down after splitting. The advice to never double or split against an ace holds whether the dealer hits or stands on soft seventeen. Table 29 assumes the dealer stands on soft seventeen. If you are playing against the lose-all rule in a casino where the dealer hits soft seventeen, then use the 10 and ace columns of table 29 with columns for other upcards from table 10 (page 48). Having to pass up those double downs and splits against lOs and aces is costly; simulation of losing all to

Lose all to a natural Win rate: $11 Std dev: $411

No

85

HOLE CARD

a natural by the dealer yields a win rate of $5 less than the benchmark. Table 30 summarizes the amendments to those parts of the appendix that deal with rules for doubling down and pair splitting against 10 or ace when you lose all to a dealer's natural. (The appendix contains strategy indexes for the high-low and halves counts for one and multiple decks and both dealer actions on soft seventeen.) No other decision is affected by whether or not the dealer takes a hole card.

Any 21 Ties Dealer Natural This section applies if the dealer's two-card twentyone counts as an ordinary twenty-one and not as a natural winner. You should be slightly more likely to hit.

Table 30 Modifications to Appendixes When You Lose All to a Natural Table Al A2 A3 A4 Bl B2 B3 B4

Double Eleven vs 10 5 4 5 4 4 3 4 3

Split

Split

Ace-Ace Ace-Ace vs 10 vs Ace -9 16 -6 24 -9 17 -6 24 -10 16 -6 23 -10 17 -6 23

86


Table 31 No Dealer Natural, S17 Player's Hand double not A-A 10-10 9-9 8-8 7-7 6-6 5-5

Dealer's Upcard 3 5 6 8 2 7 4 allowed after split spl spl spl spl spl -9 -8 4 8 6 5 -8 6 -2 -3 0 -4 -4

spl -9 2

spl spl

spl spl

spl spl

spl spl

spl spl

3-3 2-2 ace-9 ace-8 ace-7 -/db ace-7 bIace-6 ace-5 ace-4 ace-3 ace-2 hard 20 hard 19 hard 18 hard 17 hard 16

8 7 10 8 0

0 -5 -7 -3 h never split - play as total never split - play as total -2 3 0 -9 spl -4 -9 spl 3 0 8 5 4 6 5 1 1 3 -2 -6 -8 -10

1 h h h h

-3 4 7 7 7

-8

-10

hard 15

-5

hard 14 hard 13 hard 12

-3 0 3 db -8 1 h h h h

4-4

11 10 9

8** 7 6 5

h

h h h -6 -4 8 h h h h

h h h -4 -1 h h h h h

h h h

h h h

0 h h h h h h

h h h h h h h

-7 -4 0

h h h -9 -6 3 h h h h

9

h/-

h

-7

h h

h h h h h h h

of ten of eight h h h h

1

-6 -3 0

3 9 h h

h h

h h

5

-4 -1 2 db -9 0 h h h

h h

h h

8

-9

-4

h h h

spl

5

-9

5 h h h

h/h h

-9

spl

7

-7

-2

h

10

-6

9

-8

9

h h h h h

db db -6 1

-7

10

db db -9 -4 -1

db -10

A

h h h h

-10 -6 -4 -1 0

-1 db db

10

h h h h h h h

-7 -2 0 1 3

-4

9

h

1 h h h h h

No

87

HOLE CARD

Table 31 Continued Dealer's Upcard Player's 9 8 7 6 3 5 Hand 2 4 double allowed after split spl spl spl spl spl -9 -8 -7 A-A 4 8 5 10-10 6 -9 -6 3 -5 -6 -3 9-9 -2 -8 spl spl spl spl spl spl spl spl 8-8 h 5 -9 spl spl spl spl spl 7-7 h h -4 -6 6-6 -1 -8 -10 h never split - play as total of ten 5-5 h h 6 1 -4 h -1 4-4** h h 0 -4 3-3 -7 -9 spl spl 4 h -3 -5 2-2 -7 -9 spl spl 5

10

A

-8

h

h/h h

h h h

h h h

h h h

KEY db

h h/-

Stand. Double down. Hit. See elsewhere in the table for hit or stand.

spl Split. Dealer stands on soft seventeen. s17 number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. (Note that the 1 for ace-7 against ace is hit/stand, and does not say to double down.) With 4-4, splitting is better than doubling down if ** doubling is allowed after splitting.

88


The appropriate advice is shown in table 31. Table 31 is the same as table 9 (page 46) except for the 10 and ace columns. Table 31 assumes you can tie all the dealer's twocard twenty-ones by reaching twenty-one yourself in any number of cards. If your three-or-more-card twentyone ties the dealer's two-card twenty-one for only some upcards and not for others, then use part of table 31 and part of table 9. For example, you might find a casino where the dealers do not check hole cards under 10s but do check hole cards under aces. Ace-up naturals are turned up immediately, so you never get a chance to get a three-card twenty-one if the dealer shows an ace. In that case, use the 10 column from table 31 and the ace column from table 9. The other columns are the same on both tables. The ace column of table 31 applies whether the dealer stands or hits with soft seventeen. Thus for a casino in which the dealer hits soft seventeen, the 10 and ace columns of table 31 can be combined with the other columns from table 10 (page 48). When the dealer with a 10 up and ace for the second card has an ordinary twenty-one and not a natural, the win rate is better than the benchmark by $8 per hour. So this rule is quite valuable for the card counter. At the time of this writing, this rule is offered at several casinos in the Reno area.

Any 21 ties

dlr 10-up natural Win rate: $24 Std dev: $413

89

CHAPTER 8 SURRENDER Surrender is giving up half your bet and not having to play out the hand. Surrender is a good play only if there is no other way to play your hand that has an expected win of better than -0.5. In other words, since surrender means losing half a bet, any other play that on average loses less than half a bet is superior to surrender. The tables in this chapter have two columns for dealer ace. The column labeled "A-s17" is for use when the dealer stands on soft seventeen, and the column labeled "A-hI7" applies when the dealer hits soft seventeen. Whether the dealer hits or stands on soft seventeen does not matter when the dealer starts with 7 through 10. Surrender can be worthwhile if the dealer shows 8, 9, 10, or ace. When the dealer starts with 2 through 7, you should not surrender. For a complete set of strategy

90


numbers when surrender is offered, combine the appropriate table of this chapter with the 2-7 columns of the appropriate table of another chapter, such as chapter 3. The tables of this chapter break 8-8 from other player hands of sixteen because splitting 8-8 is an attractive alternative. There is no reason to separate 7-7 from other fourteens, because for upcards you might surrender against you should not split 7-7.

Early Surrender At some casinos, you have the option of early surrender; that is, you may surrender before the dealer checks the hole card. The higher the count per deck, the more likely you should be to surrender. Table 32 summarizes early-surrender decisions. Do not surrender against 2-

7. Examples: A) You get 9-7 for sixteen against ace; table 32 says always surrender. B) You get 8-8 against 8; table 32 says do not surrender. C) You get 10-4 for fourteen against 9; surrender if the count per deck is +6 or more, and do not surrender if the count per deck is less than +6. If you lose all to a natural by the dealer and thus will not split 8-8 against 10, then for 8-8 against lOuse -5, the

same number as for other sixteens against 10. Early surrender is quite profitable for the card counter, adding $19 or $23 to the benchmark win rate, with the amount depending on whether the dealer hits or stands on soft seventeen. Unfortunately, early surrender is also rare; late surrender is the common form of surrender.

91

SURRENDER

Table 32 Early Surrender Player's Hand 8

hard 17 10-6, 9-7 8-8 hard 15 hard 14 hard 13 hard 12

4 7

Dealer's Upcard 10 A-s17 A-h17 9 esr esr 5 esr esr 0 -5 esr esr 7 -2 esr 2 -2 esr -9 esr 6 0 esr -7 3 -4 8 -8

8 7 6

9 -2 -1 0 2

5 4

-6 -4 -2 0

KEY blan~

esr h17 s17

Do not surrender. Surrender. Dealer hits soft seventeen. Dealer stands on soft seventeen.

number Surrender if the count per deck equals or exceeds the number; do not surrender if the count per deck is less than the number.

Early surrender, s 17 Win rate: $39 Std dey: $402

Early surrender, hI7 Win rate: $35 Std dey: $402

92


Late Surrender At some casinos you have the option of late surrender. The higher the count per deck, the more likely you should be to surrender. Table 33 summarizes latesurrender decisions. Examples: A) You get hard thirteen against ace; table 33 says do not surrender. B) You get 10-4 for fourteen against 9; surrender if the count per deck is +6 or more, and do not surrender if the count per deck is less than +6. If you lose all to a natural by the dealer and thus will not split 8-8 against 10 or ace, then for 8-8 against 10 or ace use the same numbers as for other sixteens against 10 or ace. Late surrender is not as profitable as early surrender, of course, since you cannot surrender against a natural. But it is profitable, adding $5 to the benchmark win rate.

93

SURRENDER

Table 33 Late Surrender Player's Hand hard 17 10-6, 9-7 8-8 hard 15 hard 14 hard 13

KEY blank h 17 s17 number

Dealer's Upcard 10 A-s 17 A-h17

8

9

4

0

7

-2 0

-1

-4

7 2

0

6

3 8

2 6

4

0

Do not surrender. Dealer hits soft seventeen. Dealer stands on soft seventeen. Surrender if the count per deck equals or exceeds the number; do not surrender if the count per deck is less than the number.

Late surrender, s17 Win rate: $21 Std dev: $407

Late surrender, h17 Win rate: $17 Std dev: $408

94


Table 34 Multiple-Card Early Surrender Player's Hand

8 hard 17 10-6, 9-7 8-8 hard 15 hard 14 hard 13 hard 12

8 7 6 5 4 KEY blank esr h17 s17 number

4 7

Dealer's Upcard iO A-si7 A-hi7 9 esr 5 esr 0 -5 esr esr 7 -2 esr esr -2 esr 2 -9 0 7 -7 esr 5 -7 -4 -2 0 0 0 2 5

7 -4 -1 0 3

Do not surrender. Surrender. Dealer hits soft seventeen. Dealer stands on soft seventeen. Surrender if the count per deck equals or exceeds the number; do not surrender if the count per deck is less than the number.

Multiple-Card Surrender At a few casinos, primarily in Asia, you can surrender any number of cards. For example, if you start with

Multiple-card early surrender Win rate: $47 Std dev: $393

95

SURRENDER

Table 35 Multiple-Card Late Surrender Player's Hand 8 hard 17 10-6, 9-7 8-8 hard 15 hard 14 hard 13

Dealer's Upcard 10 A-s17 A-h17 9

4

0 7

7

2 7

-2 0 0 3 9

-1

-4

2 6

0 5

KEY

blank h 17

Do not surrender. Dealer hits soft seventeen.

s I7


number Surrender if the count per deck equals or exceeds the number; do not surrender if the count per deck is less than the number.

6-2 and hit an 8 for sixteen, you can surrender your sixteen. You are less likely to surrender a hand with a low total if you will still have the option of surrendering after one more card. The strategy numbers for multiple-card early surrender are shown in table 34. The strategy numbers for multiple-card late surrender are shown in table 35. Multiple-card late surrender Win rate: $25 Std dey: $403

96


Surrender After Doubling Down At some casinos in Asia you were allowed to surrender after doubling down. No strategy table is provided for this rule because except for seventeen against ace, the best play is simple: After doubling down, surrender all hands of sixteen or less if the dealer shows 8, 9, 10, or ace. If you have doubled down and ended up with seventeen and the dealer shows an ace, you might want to surrender. Follow the advice for seventeen against ace in the appropriate table from earlier in this chapter. When to double, if you can surrender after doubling down, is explained in the chapter on doubling down.

Surrender after doubling down Win rate: $29 Std dey: $399

Second-Chance 21 A rarely-offered form of surrender is to give up your hand plus half your bet and receive a new hand. The difference between this second-chance option and ordinary surrender is you must play a new hand for the remaining half of your bet rather than simply putting

97

SURRENDER

those chips back on your stack. If the dealer shows a card that is powerful enough to make surrender worth considering, then the dealer is showing a card you would rather not play against. Therefore, basic strategy says ignore the second-chance option. If you are counting cards and the count gets high enough, you sometimes are better off giving up half your bet to get a new hand. The situations arise when you have hard thirteen to seventeen and the dealer shows 7 to 9. How high the count per deck must be to justify giving up half your bet to get a new hand is shown in table 36. This table applies to both dealer actions on soft seventeen.

Table 36 Second-Chance 21 Player's Hand hard 17 hard 16 hard 15 hard 14 hard 13

2

3

4


9

8 4 4 8

5 5 9

3 4 5 9

10

A

KEY

blank Ignore the second-chance option. number Give up half the bet to replace the cards if the count per deck equals or exceeds the number; play out the original hand if the count per deck is less than the number.

98


CHAPTER 9 FIVE CARDS

Though this chapter is titled "Five Cards," it also covers bonuses for hands of six and seven cards. The strategy for playing them depends on the size of the bonus and the number of additional cards required to be eligible for the bonus. The tables in this chapter tell you when you should hit, given the number of additional number of cards you need to qualify for the bonus. The more cards already in your hand the closer you are to the bonus, and thus the more likely you are to hit. Of course you stop hitting and smile when you have enough cards to qualify for the bonus. Every live-game bonus I have seen has been optional, and if you accept it the hand is over and the dealer picks up your cards. (Some blackjack slot machines have a bonus that is paid automatically because no more cards will fit on the screen.)

99

FIVE CARDS

Half a Bet If the bonus is small, you might be better off refusing it. For example, Macao has a bonus of half a bet for a fivecard hand. With a bonus that small, some five-card hands are worth more when played out to maybe win, maybe push, maybe lose. Tables 37-44 are for use when the bonus is half a bet. If you happen to be offered a half-win bonus for six cards, then use tables 38-40 and 42-44 for hands of one more card; for example use table 40 for six-card hands instead of five-card hands. Tables 37-44 assume the dealer stands on soft seventeen. They also assume you can double down on any first two cards, but cannot double down on more than two cards. They also assume you lose only one bet if the dealer has a natural. The casinos that I know have offered this rule all are located in Asia. At all of them, dealers stand on soft seventeen. If you need five-card half-win tables for dealers hit soft seventeen, you can generate them yourself with Blackjack Count Analyzer. Tables 37-40 assume you cannot collect a half win if the dealer has a natural.


100


Table 37 2-Card Hands With 5-Card Half Win, 817 Dealer's Upcard Player's 3 5 6 Hand 7 8 2 4 double not allowed after split

-7 10-10 0 9-9 spl 8-8 7-7 -9 2 6-6 5-5 4-4 9 3-3 10 2-2 ace-9 10 ace-8 8 ace-7 -/db 0 ace-7 bIace-6 2 h ace-5 ace-4 h ace-3 h ace-2 h hard 20 hard 19 hard 18 hard 17 hard 16 -8 hard 15 -5 hard 14 -3 hard 13 0 hard 12 3 11 db 10 -8 9 1 8** h 7 h 6 h

A-A

5

h

-8 8 -2

-9 6 -3

spl. spl spl spl 0 -3

-10 4 -4 spl spl spl spl -5 -7 -9 5 -4

6 3 -6

5 1 -8

4 1 -10

-2 5 9 9 10

-5 -1

-8 -4 -1 1 3

db -9 -5 -1 1

2

4 6

10

A

-1

0

-6

-3

-1

6 spl spl h

-8 spl 5 h

spl h h

6* h h

spl h h

of ten of eight h h h h

h h

h h

h h h

h/-

never split - play as total never split - play as total 5 0 2 h -2 6 0 3 0 h

8 5 -2

9

3

-9

h h

h h

h

h

h

h h h

h h h h

h h

h h h h

h

h

h

h

7 10 h h h -6 -3

5 8 h h h -3 0 h

0 4 h h h -3

h

h h

h

3

-6

-10 -6 -4 -1 2 db -9 0 9 h h h

-7 -6 -3 0 db db -2

-9 -7 -4 -1 db db -4

5

3

h

9 h

-9 -7 -4 0 db db -6 1 9 h

h

h

h h

9 10 h

h h

-8 -5 4 h h h h

8

h h h h

h h h h

5 h h

8 10 h

h h

2 4 h

h h

h h h

h

h

101

FIVE CARDS

Table 37 Continued Dealer's Upcard Player's 9 6 8 7 5 4 2 3 Hand double allowed after split -1 -7 -8 A-A -9 -9 -10 -6 -3 5 4 6 8 10-10 3 -8 -6 -5 -6 -9 -2 -3 9-9 spl spl spl spl spl spl spl spl 8-8 h 7-7 -9 spl spl spl spl spl 5 h h 6-6 -1 -4 -6 -8 -10 h never split - play as total of ten 5-5 h -4 h h 6 1 h 4-4** -1 -4 h -2 -4 -7 h 2 0 3-3 h -5 1 h 1 4 -3 2-2 -1

10

A

-1

0

8* h h

3 spl h h

h h h

h h h

KEY db

Stand. Double down; if you cannot double, then hit.

h

Hit.

h/spl s17 number

See elsewhere in the table for hit or stand. Split. Dealer stands on soft seventeen. Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck below the number. (Note that the 3 for ace-7 against ace is hit/stand, and does not say to double down.) Reverse the meaning. Hit (or do not split) at a count per deck equal to or greater than the number; stand (or double or split) at a count per deck less than the number.

*

**

With 4-4, sptitting is better than doubling down if doubling is allowed after splitting.

Five-card half win Win rate: $30 Std dev: $412

102


Table 38 3-Card Hands With 5-Card Half Win, 817 Player's Hand soft soft soft soft soft soft soft soft soft

21 20 19 18 17 16 15 14 13

hard hard hard hard hard hard hard hard hard

20 19 18 17 16 15 14 13 12

2

3

4

2 h h h h h

0 h h h h h

0 h h h h h

-8

-10 -6 -4 -1

-7 -6

3

1


-6 h h h h h

-6 h h h h h

h h h h h

6 h h h h h

-9 -7 -4 0

9 h h h h

7 h h h h

9

10

A

h h h h h h

10* h h h h h h

h h h h h h

5 10 h h h

0 6 h h h

-6 -5

-3 0 4

-3

-9 -7 -4 0

KEY h

s17

Stand. Hit. Dealer stands on soft seventeen.

number Stand at a count per deck equal to or greater than the number; hit at a count per deck less than the number.

*

Reverse the meaning. Hit at a count per deck equal to or greater than the number; stand at a count per deck less than the number.

8 h h h h

103

FIVE CARDS

Table 39 4-Card Hands With 5-Card Half Win, S17 Dealer's Upcard 8 6 7 5

Player's Hand

2

3

4


h h h h h h h

h h h h h

h h h h

h

h h

h h

h h

h h

h h

-4 1 6 h

-5 0 4 9 h

-7 -1 2

-8 -3 1 5 9

-9 -3

21 20 19 18 17 16 15 14 13


20 19 18 17 16 15 14 13 12

h

7

h

h h h

h h

h h h

-5 h h

-2 h h

h

h

h h h

h h h

h

h h h h h

0

h h

5 10

h h

9

h h h h h h h

10 1* h h h

h

h h h

h

h

h

h

h

-2 h

h

h

h

h h h

h

h h

h h

Stand. IIit. Dealer stands on soft seventeen.


*

h

h h

KEY h s 17

A

Reverse the meaning. IIit at a count per deck equal to or greater than the number; stand at a count per deck less than the number.

h

h

104


Table 40 5-Card Hands With 5-Card Half Win, 817 Player's Hand soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 hard 21 hard 20 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12 II

10

KEY bon rej s17 number

*

2 rej rej bon bon bon bon bon rej rej bon bon bon bon bon bon bon bon bon bon



Dealer's Upcard 5 6 7 8 rej rej rej rej rej rej rej rej 8 rej rej 1 bon bon 10 bon bon bon bon bon bon bon bon bon bon bon bon bon rej rej rej rej rej rej rej rej 8 1 rej rej bon bon 10 bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon

9

rej rej bon bon bon bon bon rej rej bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon

10 rej 10· bon bon bon bon bon rej 10* bon bon bon bon bon bon bon bon bon bon

A

rej rej bon bon bon bon bon rej rej bon bon bon bon bon bon bon bon bon bon

Accept bonus of half a bet. Reject bonus of half a bet, and stand. Dealer stands on soft seventeen. Reject bonus and stand at a count per deck equal to or greater than the number; accept bonus at a count per deck less than the number. Reverse the meaning. Accept bonus at a count per deck equal to or greater than the number; reject bonus and stand at a count per deck less than the number.

FIVE CARDS

105

At some casinos that offer the five-card half win, dealers do not take hole cards and you lose allan doubles or splits if the dealer has a natural. At those casinos, if you get a five-card hand you can collect your half-win before you fmd out if the dealer has a natural. This book calls this "early half-win." Tables 41-44 assume you collect your half-win before the dealer takes a second card, or if one is taken, then before the dealer checks it. The columns for 2-9 are identical to tables 37-40 since the dealer cannot get a natural except with a 10 or ace up.

If you are playing the five-card half win at a casino where the dealer checks under the ace but not under the 10, then use the ace columns from tables 37-40 and the 10 columns from tables 41-44.


106


Table 41 Play of 2-Card Hands With 5-Card Early Half Win, S17 Player's Hand double not A-A 10-10 9-9 8-8 7-7 6-6

Dealer's Upcard 5 6 7 8 2 3 4 allowed after split -7 -8 -9 -9 -10 -6 -3 6 5 4 8 -4 -4 6 -8 -2 -3 0

spl -9 2

hard hard hard hard hard hard hard hard hard 1] 10 9 8** 7 6 5

20 19 ]8 17 16 15 14 ]3 12

spI spI -5

spl spl

10 8 0

8 5 -2

6 3 -6

5 1 -8

4 1 -10

2 h h h h

-2

-5 -1 2 4 6

-8 -4 -1 1 3

db

5

-8 -5 -3 0

3 db -8 1

h h h h

9

9 10

-10 -6 -4 -1 2 db

-9 0 9

h h h

spl spI

-9 -5 -1 1

h h h h h

9 -9

-7 -6 -3 0 db db

-7 -4 -1 db db

-2

-4

5 h h h

3 9 h h

spl

10

A

-1

2

h

h/h h

h/h h

h h

h h

h/8 h h h h h

-9

spl

h h -7 h h h play as total of ten play as total of eight -2 h h h 0 h h h

10

9

ace-9 ace-8 ace-7 -/db ace-7 bIace-6 ace-5 ace-4 ace-3 ace-2

spl spl

-3 0 never split never split 2 0 5 3 0 6

5-5 4-4 3-3 2-2

spl spI

9

-9 -7 -4 0 db db -6 1

9 h h

10 h h h -8 -5 4

h h h h

h h h h h h h

h h h h h h h

h h h h h h h

7 10 h h h -6 -3 8 h h h h

5

0 4 h h h 6 h h h h h h

8 h h h -3 0 h h h h h

-6 8 10 h h h

h h h h h h h

107

FIVE CARDS

Table 41 Continued Dealer's Upcard Player's 7 9 8 6 5 3 4 Hand 2 double allowed after split -1 -9 -10 -6 -7 -9 -3 A-A -8 4 5 8 6 10-10 -9 3 -5 -8 -2 9-9 -6 -6 -3 spl spl spl spl spl spl spl spl 8-8 5 h 7-7 -9 spl spl spl spl spl h h -4 -1 6-6 -6 -8 -10 h never split - play as total of ten 5-5 h -4 h h h 6 1 -1 4-4** h -7 0 -4 h -4 2 -2 3-3 h h -5 1 1 -1 4 2-2 -3

10

A

2

h

h/h h

h/h h

h h h

h h h

KEY db


h h/spl s17 number

Hit.

**


See elsewhere in the table for hit or stand. Split. Dealer stands on soft seventeen. Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. (Note that the 3 for ace-7 against ace is hit/stand, and does not say to double down.)

Five-card early half win Win rate: $36 Std dey: $411

108


Table 42 3-Card Hands With 5-Card Early Half Win, 817 Player's Hand

2

3

4


2 h h h h h

0 h h h h h

0 h h h h h

-8

-10

-5

-6

-7

-9

-9

-3 0 4

-4

-6

-1 3

-3 1

-7 -4

-7 -4

0

0

21 20 19 18 17 16 15 14 13


20 19 18 17 16 15 14 13 12


-6

-6

h h h h h

h h h h h

6

9

10

A

1* h h h h h h

h h h h h h h

h h h h h

h h h h h

h h h h h h

9 h h h h

7

5

h h h h

10 h h h

-6 0 6 h h h

KEY Stand. h Hit. s17 Dealer stands on soft seventeen. number Stand at a count per deck equal to or greater than the number; hit at a count per deck less than the number. Reverse the meaning. Hit at a count per deck equal * to or greater than the number; stand at a count per deck less than the number.

8 h h h h

109

FIVE CARDS

Table 43 4-Card Hands With 5-Card Early Half Win, 817 Player's Hand soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 soft 14 soft 13 hard 20 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12

2

3

4


9

10

A

h h h h h h h h

h h h h h h h h h

h

h h

h h h h h h h

h h h h h h h

h h h h h h h

h h h h h h h

h h h h h h h

-5 h h h h h h

-2 h h h h h h

h h h h h h h

-4

-5

-8

0 4 9 h

-3 1 5

-9 -3 0 5

h

1 6 h h

-7 -1 2 7 h

h h h

9

10

h h h h h

h h h h

h h

h h h h

KEY Stand.

h

IIit.

s 17



h h h h

110


Table 44 5-Card Hands With 5-Card Early Half Win, 817 Player's Hand soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 hard 21 hard 20 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12 11 10

KEY bon rej



4


Dealer's Upcard 5 6 7 8 rej rej rej rej rej rej rej rej rej rej 8 1 bon bon 10 bon bon bon bon bon bon bon bon bon bon bon bon bon rej rej rej rej rej rej rej rej 8 1 rej rej bon bon 10 bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon

9


10 rej -8* bon bon bon bon bon rej bon bon bon bon bon bon bon bon bon bon bon

A bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon bon

Accept bonus of half a bet. Reject bonus of half a bet, and stand. Dealer stands on soft seventeen. s17 number Reject bonus and stand at a count per deck equal to or greater than the number; accept bonus at a count per deck less than the number. Reverse the meaning. Accept bonus at a count per * deck equal to or greater than the number; reject bonus and stand at a count per deck less than the number.

111

FIVE CARDS

Table 45 3-Card Hands With 6-Card Automatic Win, 817 Player's Hand soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 soft 14 soft 13 hard 20 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12

3

4

h

h h h

h h

h

h h h h h

-8 -5 -3 0 3

-10 -6 -4 -1 2

2

-10 h h

h

-7 -6 -3 0


h h h h h

-9 -7 -4 -1

h h h h h

h h h h h

-9 -7 -4 0

9 10 h h h

A

9

10

h

h h h h h h

h h h h h h

h h h h

7 10 h h h

5 8 h h h

0 4 h h

-6 8 10 h h

h

h

-8 h h h

h

KEY

Stand. IIit. s 17 Dealer stands on soft seventeen. number Stand at a count per deck equal to or greater than the number; hit at a count per deck less than the number.

h

Six-card automatic win Win rate: $18 Std dev: $415

1 h

112


Table 46 4-Card Hands With 6-Card Automatic Win, S17 Player's Hand soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 soft 14 soft 13 hard 20 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12

2

3

4

h h h h h h

h h h h h h

h h h h h h

-8 -4 -1 2 7

-10 -6 -3 0 5

-7 -4 0 3


h h h h h h

-8 -5 -2 1

8 h h h h h

-9 -6 -2 2

9

10

A

-1 h h h h h

h h h h h h

h h h h h h

h h h h h h h

-6 5 h h h h h

9 h h h h

7 h h h h

5 h h h h

1 8 h h h

-6 8 10 h h h

KEY

h s 17

Stand. IIit. Dealer stands on soft seventeen.


113

FIVE CARDS

Table 47 5-Card Hands With 6-Card Automatic Win, S17 Player's Hand soft soft soft soft soft soft soft soft soft

21 20 19 18 17 16 15 14 13


20 19 18 17 16 15 14 13 12

Dealer's Upcard

2 h h h h h h

h h h

3

8 h

h h

8

9

h

h h h h

3

4

5

6

7

h h h

h h h h h h h h h

h h h h h h h h h

h h h h

h h h h h

h h h h h h h

h

h h h

1 6 h h h

0 4 9 h h

-1 3 7 h

-2 3 h

h h h h

0 h h h h

h

h

h

h

h

h h

8

h h h h

h

h h h

h

10 h h

h h

h h h h h

h h h h h

h h

h h

h h h h h h

7 h h h h h

h h h h

KEY h

Stand. IIit.

s 17


A h h h h


h h h

114


Table 48 2-Card Hands With db3, 6-Card Win, HI7 Player's Dealer's Upcard Hand 5 2 3 6 7 8 9 4 double not allowed after split -9 -10 -10 spl spl -7 -6 A-A -5 10-10 6 5 8 4 -2 -3 -4 -6 -1 6 -8 -9 9-9 8-8 spl spl spl spl spl spl spl spl 7-7 -8 -10 spl spl spl spl h h 6-6 -1 -3 -5 -7 h 1 h h 5-5 never split - play as total of ten 4-4** never split - play as total of eight 3-3 h 9 5 2 0 0* h h 2-2 5 h 9 2 0 h h h ace-9 10 8 6 5 4 ace-8 5 8 3 1 0 ace-7 -/db 0 -2 -6 -8 db h ace-7 hIh ace-6 2 -3 -7 -10 db h h h ace-5 0 -5 h h db h h h ace-4 h h 5 -1 -7 h h h ace-3 h 8 2 h -1 h h h ace-2 h h 9 4 2 h h h hard 20 hard 19 hard 18 hard 17 hard 16 -9 -10 9 7 5 hard 15 -5 -8 -7 -9 10 10 8 hard 14 -3 -5 -6 -7 -9 h h h hard 13 0 -3 -2 -4 -7 h h h hard 12 3 1 0 -1 -3 h h h 11 db db db db db -9 -6 -4 10 -8 -9 -10 db db -6 -4 -1 9 2 0 -1 -2 -4 4 9 h 8** h h h 9 8 h h h 7 h h h h h h h h 6 h h h h h h h h 5 h h h h h h h h

10

A

-5

-2

6* h h

2 -1 h h

h h

h h

h h h h h h h

h h h h h h h

0 4 h h h -4 4 h h h h h

-4 3 5 9 h

h 0 3 h h h h h

115

FIVE CARDS

Table 48 continued Player's 2 Hand double allowed -9 A-A 10-10 9-9 -2 spl 8-8 -9 7-7

6-6 5-5 4-4** 3-3

2-2

-2 h 3 4

Dealer's Upcard 8 9 7 5 6 4 after split -6 -5 -10 -10 spl spl -7 5 4 8 6 -8 -9 3 -3 -5 -6 -8 spl spl spl spl spl spl spl 5 h spl spl spl spl spl h h -4 -6 -8 -10 h never split - play as total of ten h h h 3 0 h 8 h 0 -2 -4 -6 spl h h h 0 -2 -4 -6 -5 3

10

A

-5

-2

8* h h

1 -1 h h

h h h

h h h

KEY -/db db

h hI7 h/spl

Stand. Do not hit. See the -/db row for the number saying whether to stand or double down. Double down; if you cannot double, then hit. Hit. Dealer hits soft seventeen. Do not double. See the h/- row for the number saying whether to hit or to stand. Split.

number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. (Note that the 3 for ace-7 against ace is hit/stand, and does not say to double down.)

*

**

Reverse the meaning. Hit (or do not split) at a count per deck equal to or greater than the number; stand (or double or split) at a count per deck less than the number. With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

116


Table 49 3-Card Hands With db3, 6-Card Win, HI7 Player's Hand soft 21 soft 20 soft 19 soft 18 -/db soft 18 hIsoft 17 soft 16 soft 15 soft 14 soft 13 hard hard hard hard hard hard hard hard hard

20 19 18 17 16 15 14 13 12

11 10 9 8 7 KEY

2

3

10 8 0 -9

8

5 h h h

h

-9

-5 -3 0

3 db -7 3 h h

5 -2 -10 0

h h h h

4


9

10

A

h h h h h h

h h h h h h h

h h h h h h h

5 4 6 0 1 3 -8 db -6 -10 s/db s/db -6 -10 -3 -6 2 -1 2 -1 8 5 9 3 6 h 7

h h h h h

h/-8 h h h h h

9 10 h h h

7 10 h h h

5 8 h

0 4 h

h

h

h

h

-8 -5 5 h h

-5

-2

-2

-2 9

0 h

h h

h h

7 h h h

-10 -7 -5 -2 1

-8 -6 -3 0

-9 -7 -4 -1

db -8 0

db -9 0

h h

h

db db -2 9 h

h

-9 -7 -3

db db -4 8 h

h

Stand. -/db

Do not hit. See the -/db row for the number saying whether to stand or double down.

db

Double down; if you cannot double, then hit.

h

Hit.

hI7


h/-

Do not double. See the h1- row for the number saying whether to hit or to stand.

number

Stand or double at a count per deck equal to or greater than the number; hit (or do not double) at a count per deck less than the number.

-4 3

5 9 h h 1 4 h h h

117

FIVE CARDS

Table 50 4-Card Hands With 6-Card Win, HI7 Player's Hand soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 soft 14 hard hard hard hard hard hard hard hard hard 11 10 9 8

20 19 18 17 16 15 14 13 12

2

3

4

10

8

6

8

5 h h h h h

3 h 10 h h h

h h h h h

-9 -5 -2 1 6

-10

h h h

h h h h

h

-6 -3 0 4

Dealer's Upcard 7 8 6 5 5 1 h 7 h h h

9

10

A

h h h h h h

0 h h h h h

1 8 h h h

-4 3 7 h h h

h h

h h

h

h h

4 0 h 5 h h h

-1 h h h h

h h h h h

h h h h h

7 h h h h

5 h

-4 0

9 h h h h

h h h h

-7 -4 -1 3

-8 -6 -2 1

h h h

h h h

h h h

h h h

h

h

h

h

-8

h h h

h h h h

h

KEY Stand.

h IIit. h 17 Dealer hits soft seventeen. number Stand or double at a count per deck equal to or greater than the number; hit (or do not double) at a count per deck less than the number.

118


Table 51 5-Card Hands With 6-Card Win, Ht7 Player's Hand soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15

hard hard hard hard hard hard hard hard hard 11 10

20 19 18 17 16 15 14 13 12

2 h h h h h h h

3 8 h h h

h h

h h h h h h h

4 h h h h h h h

1 6 h h h h h

0 4 9 h h h h

3

Dealer's Upcard 5 6 7 8 h h h h h h h h h h h h h h h h h h h h h h h h h h h h

0 h h h h h h h

-2 3 7 h h h h

-3 2 7 h h h h

h h h h h h h

9

h h h h h h h

10 h h h h h h h

A h h h h h h h

h h h h h h h h

h h h h h h h h

9 h h h h h h h

KEY

Stand. h IIit. h 17 Dealer hits soft seventeen. number Stand at a count per deck equal to or greater than the number; hit at a count per deck less than the number.

119

FIVE CARDS

Automatic Win Tables 45-51 cover the situation of six cards is an automatic winner of the same amount as you have bet. In this case it is always worthwhile accepting the bonus - you should never turn it down. Note that it is worthwhile hitting soft twenty-one if one more card will give you an automatic winner. A sure winner is worth more than a hand that will probably win but might push. Tables 45-51 assume the automatic winner applies to six-card hands. If you happen to be offered an automatic win for five cards, then use the tables 45-51 for hands of one less card; for example use table 47 for fourcard hands instead of five-card hands. Likewise if you happen to be offered an automatic winner for seven-card hands, then use tables 45-51 for hands of one more card. Tables 45-51 also assume you lose only one bet if the dealer has a natural. Tables 45-47 assume the dealer stands on soft seventeen, and they assume you can double down on any first two cards. Tables 48-51 assume the dealer hits soft seventeen and also assume you can double down on two or three cards. Las Vegas Club in Las Vegas (at this writing) has these rules, and allows double down on two cards after splitting. The numbers for soft eighteen in table 48 are for stand versus double. If the count per deck is below the number in the table, then stand; if the count per deck is equal to or greater than the number in the table, then double. Soft eighteen is more complicated if: you have it in three cards, you can double down on three cards, and a six-card hand is an automatic winner. Therefore, table (text continues on page 124)

120


Table 52 2-Card Hands With Winning 5-Card 21 Pays 2-1, S17 Dealer's Upcard Player's 9 6 7 8 5 2 3 Hand 4 double not allowed after split -6 -6 -10 -10 spl spI spI -8 A-A 6 4 5 10-10 8 -3 -4 -4 6 -8 -9 0 -2 9-9 spl spl spl spl spI spI spI spl 8-8 h h -9 spl spl spI spl spl 7-7 h -3 -5 -7 h h 0 2 6-6 never split - play as total of ten 5-5 never split - play as total of eight 4-4 -1 -4 8 h h 0 8 4 3-3 -6 0 h h 0 -2 2-2 7 4 6 5 4 10 8 ace-9 3 1 1 5 8 ace-8 -8 -10 h -6 0 -2 ace-7 -/db h ace-7 hIh h -3 -7 -10 db h ace-6 1 -2 -6 db h h h 4 ace-5 h 0 -3 -8 h h h ace-4 h 7 -1 -4 h h h 7 ace-3 h 1 h h -1 h h 3 1 ace-2 7

hard hard hard hard hard hard hard hard hard 11 10 9 8** 7 6 5

20 19 18 17 16 15 14 13 12

10

A

-6

-2

6* h h

spl

h h

h h

h h h h h h h

h/2 h h h h h

3 h h

-6

-8 -5 -3 0 3

-10

db

db

db

-8

-9

1 h h h h

0 9 h h h

-10 -2 5 h h h

-6 -4 -1 2

-7

-6 -3 0

-9 -7 -4 -1

-9 -7 -4

0

9 10 h h h

db db

db db

-9 -5

7 10 h h h -6 -3

-4 3

-6 1 9 h h

4 h h h h

h h h h

9 h

h

8

5 8 h h h

-4 0 h h h h h

0 4 h h h -4 5 h h h h h

8 10 h h h 1 4 h h h h h

121

FIVE CARDS

Table 52 Continued Dealer's Upcard Player's 7 8 6 5 3 4 2 Hand double allowed after split -6 -10 -10 spl spl spl -8 A-A 4 6 5 8 10-10 3 -8 -5 -2 -3 -6 -7 9-9 8-8 7-7

spl

6-6 5-5

-1

4-4**

h 0 0

3-3 2-2

-9

spl spl -4

spl spl -6

spl spl -8

spl spl -10

spl spl h

never split - play as total 0 h 3 0 spl spl -5 -7 -5 -7 -10 -6 -3

8 -1

spl 5 h

9

10

A

-6

-6

-2

8* h h

spl h h

h h h

h h h

-9 spl h h

of ten h h 9 h

h h

3

KEY Stand. -/db


db


h h/-

Hit.

spl

Split.

s17


number

Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. (Note that the 3 for ace-7 against ace is hit/stand, and does not say to double down.)

*

Reverse the meaning. Hit (or do not split) at a count per deck equal to or greater than the number; stand (or double or split) at a count per deck less than the number.

**


Do not double. See the h/- row for the number saying whether to hit or to stand.

122


Table 53 3-Card Hands With Winning 5-Card 21 Pays 2-1, S17 Player's Hand soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 soft 14 soft 13 hard 20 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12

2

3

4

-10

-10 h h h

-10 h h h h h

h h h h h

-8 -5 -3 0 3

h h


9

10

A

h h h h h

h h h h h

h h h h h

-8 h h h h h

h h h h h h

h h h h h h

6 h h h h h

-9 -7 -4 0

9 h h h h

7 h h h h

5 10 h h h

0 5 h h h

-6 8 h h h h

-10

-6 -4 -1 2

-7 -6 -3 0

-9 -7 -4 -1

KEY

h s 17

Stand. I1it. Dealer stands on soft seventeen.


123

FIVE CARDS

Table 54 4-Card Hands With Winning 5-Card 21 Pays 2-1, 817 Player's Hand soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 soft 14 soft 13 hard 20 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12


2

3

4

h h h h h h

h h h h h h

h h h h h h

-6 h h h h h

-6 h h h h h

h h h h h

-4 -1 0 4 8

-6 -3 0 2 6

-7 -4 -2 0 4

-8 -5 -3 0 3

-9 -6 -3 0 4

-7 h h h h h

9

10

A

h h h h h h

h h h h h h

h h h h h h

h h h h h h

-8 h h h h h

h h h h h

10 h h h h

0 h h h h h

KEY

Stand. h IIit. s 17 Dealer stands on soft seventeen. number Stand at a count per deck equal to or greater than the number; hit at a count per deck less than the number.

124


49 has two rows for soft eighteen. One row is for the double-down decision, and one row is for the hit/stand decision. Always check the double-down row first; if your count per deck is high enough to justify a double down, then do not bother with the hit/stand row.

Double Pay on Five-Card 21 Tables 52-54 cover the situation where five cards is an automatic 2: 1 winner only if you get exactly twentyone and the dealer does not also get twenty-one. This bonus is less valuable than if you were to get the bonus no matter what the total points in your hand, and no matter whether that total was a winner. When the number of cards required for the bonus is five and you must get exactly twenty-one and the total must win, the bonus adds only about $1 per hour to the benchmark. This bonus has been offered at a few tables in Atlantic City since 1991.

Double pay on five-card 21 Win rate: $17 Std dev: $415

125

CHAPTER 10 6-7-8 ETC This chapter covers bonuses for 6-7-8 and three 7s. Each table in this chapter is divided into several sections. The reason is there are several ways that the bonus can interact with surrender. You might be paid the bonus even if the dealer turns out to have a natural on the hand, or the bonus might be paid only when the dealer does not have a natural. And there are both early and late surrender. All of the tables in this chapter assume you are paid the bonus if the dealer achieves twenty-one in more than two cards.

126


6-7-8 The next few sections apply to bonuses for 6-7-8. If the bonus applies only to one suit, such as spades, the numbers in the sections for 6-7-8 suited apply. You simply use these numbers when your first two cards are of the appropriate suit. Tables 55-58 are for use when the dealer stands on soft seventeen; tables 59-62 are parallel and apply when the dealer hits soft seventeen.

Double Pay for 6-7-8 Double for Mixed Suits In 1990 The Red Lion in Elko paid double for any 67-8; the cards did not have to be of the same suit. This made hitting 6-7, 6-8, and 7-8 more attractive. The appropriate strategy numbers defining where hitting becomes more attractive than standing are shown in table 55. Red Lion did not offer surrender. However, table 55 also indicates when it is better to hit than to earlysurrender, and when it is better to hit than to latesurrender. Note that hi-means do not surrender; instead use the hit-stand part of the table.

Double for any 6-7-8 Win rate: $21 Std dev: $417

6-7-8

127

ETC

Table 55·

Any 6-7-8 Pays Double, S17 Dealer's Upcard Player's 7 8 5 6 3 2 4 Hand hit/stand h -5 -4 h -3 any 7-8 -2 -1 -2 h h 0 any 6-8 1 -1 -2 b 0 0 h 3 1 5 any 6-7 early surrender, assuming 6-7-8 beats a natural b h h/- h/- h/- b/- h/any 7-8 h h b/- h/- h/- h/- h/any 6-8 b/- h/- h/- h/- h/h h any 6-7 assuming 6-7-8 loses to a natural early surrender, any 7-8 h/- h/- h/- h/- h/h h h/- h/- h/- h/- h/h h any 6-8 b/- h/- h/- h/- h/any 6-7 h h late surrender, assuming 6-7-8 beats a natural h/- h/- h/- h/- h/any 7-8 h h any 6-8 h/- h/- h/- h/- h/h h h/- h/- h/- h/- h/any 6-7 h h late surrender, assuming 6-7-8 loses to a natural any 7-8 h/- h/- h/- h/- h/b h any 6-8 h/- h/- h/- h/- h/h b h/- h/- h/- b/- h/any 6-7 h h

9

10

A

h h b

h h b

h h b

7 h h

3 8 h

-1

7 h h

2

6 h

-6 -7 -7

7 h h

6 b h

h h h

7 h h

5 h b

7 h b

0 3

KEY h

flit.

hiSee the first three lines of the table for hit or stand. Dealer stands on soft seventeen. s 17 number Stand (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; bit if tbe count per deck is less than the number.

128


Double For Same Suit Many casinos have offered double pay for 6-7-8 of the same suit. That is not nearly as valuable as double pay for any 6-7-8. And, since chances of catching the card needed for the bonus are only 1/4 as high as if mixed suits would earn the bonus, hitting suited 6-7 or 6-8 or 7-8 is not as attractive as if any 6-7-8 paid double. Table 56 indicates when it is better to hit than to stand, when it is better to hit than to early-surrender, and when it is better to hit than to late-surrender.

Double for suited 6-7-8 Win rate: $16 Std dey: $415

6-7-8

ETC

129

Table 56 6-7-8 Suited Pays Double, S17 Dealer's Upcard Player's 9 10 A 45678 Hand 2 3 hit/stand 10 7 h -6 -7 -8 h h 7-8 ss -4-5 h h h -4 -6 -6 h h 6-8 ss -2-3 h h h -1 -3 -3 h h 6-7 ss 1 0 early surrender, assuming 6-7-8 suited beats a natural 7-8 ss h/- h/- h/- h/- h/h 8 4 0 -9 6-8 ss h/- h/- h/- h/- h/h h 8 1 -9 6-7 ss hI- hI- hI- hI- hIh h h 5 -7 early surrender, assuming 6-7-8 suited loses to a natural 7-8 ss hI- hI- hI- hi- hIh 8 4 -1 -1 0 h/- h/- h/- h/- h/h h 8 1 -9 6-8 ss 6-7 ss hI- hi- hi- hi- hIh h h 5 -7 late surrender, assuming 6-7-8 suited beats a natural 7-8 ss h/- h/- h/- h/- h/h 8 4 1 3 h/- h/- h/- h/- h/h h 8 5 8 6-8 ss hi- hi- hi- hI- hih h h 10 h 6-7 ss late surrender, assuming 6-7-8 suited loses to a natural 7-8 ss h/- h/- h/- h/- h/h 8 4 1 3 h/- h/- h/- h/- h/h h 8 5 8 6-8 ss 6-7 ss hI- hi- hi- hi- hih h h 10 h

KEY h

lIit.

hiss

See the fIrst three lines of the table for hit or stand. Same suit.

s 17


number Stand (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

130


Triple Pay For 6-7-8 Triple for Mixed Suits I know of no casino that has offered triple pay for 67-8 of mixed suits. However, should you encounter this rule, table 57 indicates when it is better to hit than to stand, when it is better to hit than to early-surrender, and when it is better to hit than to late-surrender.

Triple for any 6-7-8 Win rate: $25 Std dev: $418

6-7-8

131

ETC

Table 57 Any 6-7-8 Pays Triple, S17 Dealer's Upcard Player's 8 6 5 7 3 Hand 2 4 hit/stand -1 h h 0 -1 any 7-8 2 0 0 1 h h 2 any 6-8 5 4 h 5 4 h any 6-7 10 7 6 early surrender, assuming 6-7-8 beats a natural h/- h/- h/- h/- h/h h any 7-8 h/- h/- h/- h/- h/h any 6-8 h h/- h/- h/- h/- h/h h any 6-7 early surrender, assuming 6-7-8 loses to a natural any 7-8 h/- h/- h/- h/- h/h h any 6-8 h/- h/- h/- h/- h/h h any 6-7 h/- h/- h/- h/- h/- h h late surrender, assuming 6-7-8 beats a natural any 7-8 h/- h/- h/- h/- h/h h any 6-8 h/- h/- h/- h/- h/b h any 6-7 h/- h/- h/- b/- b/h h late surrender, assuming 6-7-8 loses to a natural any 7-8 b h/- h/- b/- h/- h/h any 6-8 b h/- h/- h/- h/- b/h any 6-7 h/- h/- b/- h/- h/b h

9

10

A

h h h

h h h

h h h

h h h

7 h h

4 10 h

h h h

6 h h

-1 0

h h b

h h h

h b h

h b h

9

h h

h h h

-2

KEY IIit. h/See the fIrst three lines of tbe tabIe for hit or stand. s 17 Dealer stands on soft seventeen. number Stand (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

h

132


Triple for Same Suit Triple pay for 6-7-8 of the same suit has been a rule in Macao and a few other places in Asia. The value of this rule to a card counter is minimal. Table 58 indicates when it is better to hit than to stand, when it is better to hit than to early-surrender, and when it is better to hit than to late-surrender.


Triple for suited 6-7-8 Win rate: $16 Std dev: $416

6-7-8

ETC

133

Table 58 6-7-8 Suited Pays Triple, S17 Dealer's Upcard Player's 10 9 6 7 8 5 4 3 Hand 2 hit/stand 9 h -7 h -6 h -5 -4 7-8 ss -3 h h h -5 -5 h -3 -1 -2 6-8 ss h h -2 h h -2 0 0 6-7 ss 2 early surrender, assuming 6-7-8 suited beats a natural 0 5 9 h 7-8 ss h/- b/- h/- h/- h/10 3 h h h/- b/- h/- h/- h/6-8 ss 7 h h h h/- b/- h/- h/- h/6-7 ss early surrender, assuming 6-7-8 suited loses to a natural 0 9 5 7-8 ss h/- h/- h/- h/- h/h 10 2 h h h/- h/- h/- h/- h/6-8 ss h 7 6-7 ss h h h/- h/- h/- h/- h/late surrender, assuming 6-7-8 suited beats a natural 7-8 ss h/- b/- h/- h/- h/5 3 h 9 7 h 10 6-8 ss h/- h/- h/- h/- h/h 6-7 ss h/- h/- h/- h/- h/h h h h late surrender, assuming 6-7-8 suited loses to a natural 7-8 ss h/- h/- h/- h/- h/h 9 5 2 h 10 7 6-8 ss h/- h/- h/- h/- h/h h/- h/- h/- h/- h/6-7 ss h h h h

A h h h -7 -9 -7 -9 -9 -7 7 h h 4 h h

KEY

b

flit.

h/ss s17 number

See the first three lines of the table for hit or stand. Same suit. Dealer stands on soft seventeen. Stand (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

134


Table 59 Any 6-7-8 Pays Double, HI7 Dealer's Upcard Player's 7 8 3 5 6 2 4 Hand hit/stand -4 -5 h h any 7-8 -2 -3 -1 -1 -2 -2 h h any 6-8 1 0 0 0 h h any 6-7 5 3 1 early surrender, assuming 6-7-8 beats a natural h any 7-8 h/- h/- h/- h/- h/h any 6-8 h/- h/- h/- h/- h/h h any 6-7 h/- h/- h/- h/- h/h h early surrender, assuming 6-7-8 loses to a natural any 7-8 h h/- h/- h/- h/- h/h any 6-8 h/- h/- h/- h/- h/h h any 6-7 h/- h/- h/- h/- h/h h late surrender, assuming 6-7-8 beats a natural any 7-8 h/- h/- h/- h/- h/h h any 6-8 h/- h/- h/- h/- h/h h any 6-7 h/- h/- h/- h/- h/h h late surrender, assuming 6-7-8 loses to a natural any 7-8 h/- h/- h/- h/- h/h h h/- h/- h/- h/- h/any 6-8 h h any 6-7 h/- h/- h/- h/- h/h h

9

10

A

h h h

h h h

h h h

7 h h

3 8 h

-1 0

7 h h

2 6 h

-6 -7 -7

7 h h

6 h h

h h h

7 h h

5 h h

7 h h

3

KEY

h h/-

h!7

Hit. See the fIrst three lines of the table for hit or stand. Dealer hits soft seventeen.

number Stand (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

6-7-8

ETC

135

Table 60 6-7-8 Suited Pays Double, HI7 Dealer's Upcard Player's 6 8 9 10 A 5 7 Hand 2 3 4 hit/stand h -8 10 h 7-8 ss -7 7 -5 -6 -4 h 6-8 ss h h h h -4 -6 -6 -2 -3 h h h h h -3 -3 0 6-7 ss 1 -1 h early surrender, assuming 6-7-8 suited beats a natural 7-8 ss 8 4 0 -9 h/- h/- h/- h/- h/h 6-8 ss h/- h/- h/- h/- h/h h 8 1 -9 h h h 5 -7 6-7 ss h/- h/- h/- h/- h/early surrender, assuming 6-7-8 suited loses to a natural 7-8 ss h/- h/- h/- h/- h/h 8 4 -1 -10 6-8 ss h/- h/- h/- h/- h/h h 8 1 -9 6-7 ss hi- hi- hi- hi- hih h h 5 -7 late surrender, assuming 6-7-8 suited beats a natural 7-8 ss h/- h/- h/- h/- h/h 8 4 1 3 6-8 ss h/- h/- h/- h/- h/h h 8 5 8 6-7 ss hi- hi- hi- hi- hih h h 10 h late surrender, assuming 6-7-8 suited loses to a natural 7-8 ss h/- h/- h/- h/- h/h 8 4 1 3 6-8 ss h/- h/- h/- h/- h/h h 8 5 8 6-7 ss hi- hi- hi- hi- hih h h 10 h KEY

h hi-

IIit. See the first three lines of the tabIe for hit or stand.

h 17


ss Same suit. number Stand (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

136


Table 61 Any 6-7-8 Pays Triple, H17 Dealer's Upcard Player's 8 5 Hand 3 6 7 2 4 hit/stand -1 -1 0 h any 7-8 0 h 2 0 4 2 1 h any 6-8 5 h 6 4 5 h 10 7 h any 6-7 early surrender, assuming 6-7-8 beats a natural any 7-8 h h/- h/- h/- h/- h/h h any 6-8 h/- h/- h/- h/- h/h any 6-7 h h h/- h/- h/- h/- h/early surrender, assuming 6-7-8 loses to a natural any 7-8 h/- h/- h/- h/- h/h h any 6-8 h/- h/- h/- h/- h/h h any 6-7 h/- h/- h/- h/- h/h h late surrender, assuming 6-7-8 beats a natural any 7-8 h/- h/- h/- h/- h/h h any 6-8 h/- h/- h/- h/- h/h h any 6-7 h/- h/- h/- h/- h/h h late surrender, assuming 6-7-8 loses to a natural any 7-8 h/- h/- h/- h/- h/h h any 6-8 h/- h/- h/- h/- h/h h any 6-7 h/- h/- h/- b/- h/h h KEY h

9

10

A

h h h

h h h

h h h

h h h

7 h h

10 h

h b h

6 h h

-2 -1 0

h h h

h h h

h h h

h h h

9

h h

h h h

4

Hit. See the fIrst three lines of the table for hit or stand. hI7 Dealer hits soft seventeen. number Stand (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number. h/-

6-7-8

137

ETC

Table 62 6-7-8 Suited Pays Triple, HI7 Player's Hand hit/stand


9

10

A

9 h h h -7 -6 -5 h h h -5 h -5 -3 h h h h -2 -2 0 0 2 early surrender, assuming 6-7-8 suited beats a natural 5 0 9 h h/- h/- h/- h/- h/7-8 ss 3 10 h h h/- h/- h/- h/- h/6-8 ss 7 h h h h/- h/- h/- h/- h/6-7 ss early surrender, assuming 6-7-8 suited loses to a natural 7-8 ss h/- h/- h/- h/-- h/h 9 5 0 6-8 ss h/- h/- h/- h/- h/h h 10 2 6-7 ss hi- hi- hi- hi- hih h h 7 late surrender, assuming 6-7-8 suited beats a natural 7-8 ss h/- h/- h/- h/- h/h 9 5 3 6-8 ss hi- hi- hi- hi- hih h 10 7 6-7 ss h/- h/- h/- h/- h/h h h h late surrender, assuming 6-7-8 suited loses to a natural 7-8 ss h/- h/- h/- h/- h/h 9 5 2 6-8 ss hi- hi- hi- hi- hih h 10 7 6-7 ss h/- h/- h/- h/- h/h h h h

h h h

7-8 ss 6-8 ss 6-7 ss

2 -3 -1

3

4

-4

-2

-7 -9 -7 -9 -9 -7 7 h h 4 h h

KEY

h h/-

lIit. See the fIrst three lines of the table for hit or stand.

h 17


ss Same suit. number Stand (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

138


7-7-7 This section applies when you get a bonus for 7-7-7. It assumes that you are not allowed to double after splits. Table 63 gives the strategy appropriate for splitting a pair of 7s when 7-7-7 (any mixture of suits) pays double, and table 64 applies when 7-7-7 pays triple. Tables 63-64 are for use when the dealer stands on soft seventeen; tables 65-66 are parallel and apply when the dealer hits soft seventeen. (text continues on page 142)

Table 63 7-7-7 Pays Double, S17 Player's Dealer's Upcard Hand 5 6 7 2 3 8 4 split -7 -10 spl any 7-7 -2 -4 -6 h early surrender, assuming 7-7-7 beats a natural h/spl h/spl h/spl h/spl h/spl spl any 7-7 h early surrender, assuming 7-7-7 loses to a natural any 7-7 h/spl h/spl h/spl h/spl h/spl spl h late surrender, assuming 7-7-7 beats a natural any 7-7 h/spl h/spl h/spl h/spl h/spl spl h late surrender, assuming 7-7-7 loses to a natural h/spl h/spl h/spl h/spl h/spl spl any 7-7 h

9

10

A

h

h

h

h

7

0

h

5

-4

h

h

h

h

10

h

KEY

h h/spl

Hit. See the fIrst line of the table for hit or split.

spl Split. s17 Dealer stands on soft seventeen. number Split (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

6-7-8

139

ETC

Table 64 7-7-7 Pays Triple, S17 Player's Dealer's Upcard Hand 2 3 4 5 6 7 8 split any 7-7 0 -I -3 -4 -7 -3 h early surrender, assuming 7-7-7 beats a natural any 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h early surrender, assuming 7-7-7 loses to a natural any 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h late surrender, assuming 7-7-7 beats a natural any 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h late surrender, assuming 7-7-7 loses to a natural any 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h

9

10

A

h

h

h

h

h

8

h

h

0

h

h

h

h

h

h

KEY h h/spl

Hit. See the fIrst line of the table for hit or split. sl7 Dealer stands on soft seventeen. number Split (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

Double for 7-7-7 Win rate: $16 Std dey: $415

Triple for 7-7-7 Win rate: $16 Std dey: $416

140


Table 65 7-7-7 Pays Double, HI7 Player's Dealer's Upcard Hand 2 3 4 5 6 7 8 split any 7-7 -2 -4 -6 -7 -10 spl h early surrender, assuming 7-7-7 beats a natural any 7-7 h/spl h/spl h/spl h/spl h/spl spl h early surrender, assuming 7-7-7 loses to a natural any 7-7 h/spl h/spl h/spl h/spl h/spl spl h late surrender, assuming 7-7-7 beats a natural any 7-7 h/spl h/spl h/spl h/spl h/spl spl h late surrender, assuming 7-7-7 loses to a natural any 7-7 h/spl h/spl h/spl h/spl h/spl spl h

9

10

A

h

h

h

h

7

0

h

5

-4

h

h

h

h

10

h

KEY h h/spl

Hit. See the fIrst line of the table for hit or split. Dealer hits soft seventeen. hI7 spl Split. number Split (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

6-7-8

141

ETC

Table 66 7-7-7 Pays Triple, Ht7 Player's Hand split

2

3

4

Dealer's Upcard 5

6

7

8

9

10

A

any 7-7

0

-1

-3

-4

-7

-3

h

h

h

h

h

h

8

h

h

0

h

h

h

h

h

h

early surrender, assuming 7-7-7 beats a natural any 7-7

h/spl h/spl h/spl h/spl h/spl h/spl

h

early surrender, assuming 7-7-7 loses to a natural any 7-7


h

late surrender, assuming 7-7-7 beats a natural any 7-7


h

late surrender, assuming 7-7-7 loses to a natural any 7-7


h

KEY h

Hit.

h/spl

See the frrst line of the table for hit or split.

h17


number Split (against 2 through 6) or surrender (against 8, 9, 10, or ace) if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

142


CHAPTER 11 OVER/UNDER 13

The over/under is two side bets. You can bet that your first two cards will total over thirteen, or that they will total under thirteen. Aces count one, and thirteen loses. This innovation started in 1988 at Caesars Tahoe and hopefully will continue to spread.

Basic Strategy Basic strategy says ignore the over/under side bets. You can make money on them only by counting cards. Both the over and under are horrible bets right after a shuflle. The over loses at the rate of more than 6% and the under loses at a rate of more than 10%.

143

OVER/UNDER

Single Deck What makes the over/under beatable is volatility. As a card is dealt out and set aside not to be reused until after the next shufile, the change in the casino's edge is much greater for the over/under bets than for regular blackjack. Here is an example showing how strong the over bet is. With a freshly-shuffled single deck, there are 1326 two-card combinations, of which 618 exceed thirteen and win while 708 lose, for a casino edge of 6.8%. Removing two small cards leaves 50 cards from which can be dealt 1225 two-card combinations, of which 618 win and 607 lose, for a player edge of 0.9%. Removing a third small card leaves 49 cards from which can be dealt 1176 two-card combinations, of which 618 win and 558 lose, for a player edge of 5.1%. Removing a fourth small card leaves 48 cards from which can be dealt 1128 twocard combinations, of which 618 win and 510 lose, for a player edge of 9.6%. Removal of one card per deck can make a difference of about 4% in expected value of the over bet. The most effect any card has in regular blackjack is about 0.8% for removal of a 5. You do not fmd profitable over or under bets very often, but some of the bets you make have very high expected values. When the Comstock in Reno first introduced the over/under bets at single and double deck, it had a $100 maximum on those bets and was casual about $100 bets. I made mostly flat $100 bets and played basic strategy, and when the crush count (explained later in this chapter) told me I had an edge on the over or the under, I made the appropriate side bet with another $100 chip. One time after I placed my bets on two hands, each with an over bet of $100, but before

144


the dealer started to distribute cards, the action was halted for replenishing the dealer's chip supply. While waiting, I calculated my expected win on the coming hand. I had a running crush count of +9 with about half a deck remaining to be dealt. At 2% per count per deck, my edge was 30% on my over bets! A shufile would have cost me $60 in expected value! Mentally I shouted "Please don't shuffle! Please don't shuffle!" Fortunately the dealer did not shuffle; she dealt me two hands over thirteen. The single-deck over/under simulation results reported here are for benchmark rules plus over/under. At the Comstock during its big over/under giveaway, dealers hit soft seventeen and players could double down only on ten or eleven. Those two rules variations reduced the win rate to $62. For single or double deck I suggest playing basic strategy and not deviating from it based on the count, except to make profitable over/under bets.

Over/under, high-low, one deck Win rate: $78 Std dev: $544

Over/under, crush count, one deck Win rate: $135 Std dev: $624

145

OVER/UNDER

You will note from the simulation results that the standard deviation is higher when you occasionally bet the over or under. This increased risk means you may want to decrease your bet size somewhat or take a larger bankroll to handle the larger fluctuations you will encounter.

Counting Cards Commonly-used counting systems are not well suited to the over/under decision. For the high-low, bet the under at counts below -6 and bet the over at counts above 5. Those numbers apply to all numbers of decks. Counts that far from zero do not occur often, reflecting the fact that the high-low is not a good counting system to use for the over/under. Counting systems that are good for the over are also good for the under. The pioneering publication explaining how to get an edge on the over/under side bets is "The Over/Under Report," an excellent paper by Arnold Snyder. I have checked the numbers in this paper, including calculating all the correlation coefficients for one to eight decks, and I found no mistakes. I am not going to reproduce Snyder's work here. But I will introduce the counting system that I prefer to use for the over/under side bets, a system not mentioned in Snyder's paper. It was called the crush count by Jake Smallwood in the March 1990 issue of Blac/ifack Monthly. The crush count has a correlation coefficient of 0.97 for

146


Table 67 Crush Count Card

Count

Ace,2

+2 +1

3 4,5,6,7,8 9,10,J,Q,K

0 -1

the over and -0.95 for the under. It is a level 2 count, meaning that the values you assign to the cards are integers from -2 to +2. Snyder's best level 2 count is slightly more efficient at 0.98 for the over and -0.96 for the Wlder. I prefer the crush COWlt because it assigns the value of zero to more cards, making it easier to use. The crush count is shown in table 67. It counts +2 for A and 2, +1 for 3, and -1 for 9-10. Cards 4-5-6-7-8 count

Over/under, six decks, high-low COWlt Win rate: $32 Std dev: $455

Over/under, six decks, crush count Win rate: $63 Std dev: $549

Over/under, six decks, crush for over/under only high-low for other decisions Win rate: $66 Std dev: $469

OVER/UNDER

147

zero. Bet the over on crush counts per deck of 3 or more, and bet the under on crush counts per deck of -5 or lower. Those numbers apply to all numbers of decks. (A reminder: -6 is lower than -5; -4 is not lower than -5.) Each count per deck of the crush count changes your expectation on over and under bets by about 2.1 %. The break-even count per deck on the over is 3.1 for single and double deck, and 3.0 for more than two decks. The break-even count per deck on the under is -4.6 for all numbers of decks. You can get a healthy edge at the over/under, if you can play without the dealer's shuftling up on you. The simulation results reported in the rectangles are for the benchmark betting scheme, including a matching bet on the over or under if justified. Better penetration means a higher expected win rate, probably more so than with any other blackjack rule.

148


CHAPTER 12 HALVES COUNT

The counting system introduced in chapter 3 is called the high-low system. The counting system introduced in this section will be referred to as counting halves. Both counting systems have the same efficiency with respect to insurance and each requires the same amount of memorizing from tables. Counting halves is more difficult to do, but will result in a slightly higher expected-win rate. Counting halves means assigning values to cards as shown in table 68. (If you prefer to work with integers you can double all these card values, and also double the strategy numbers in tables 69 and 70.) A full deck contains 20 plus and 20 minus points in the high-low system, and 22 plus and 22 minus points in the halves system. The halves system finds more situations of advantage than the high-low system does. The

149

HALVES COUNT

halves count is a very close approximation of the precise value of each card as given on page 162 of the 1962 edition of Thorp's Beat the Dealer. With typical rules, the dealer's advantage disappears when either high-low or halves count per deck rises to +1. According to Thorp's book, each additional count per deck gives you an extra 0.56% of advantage in the high-low count, with a standard deviation of 0.22%. Thus the value of a high-low count per deck plus or minus one standard deviation is 0.34% to 0.78%. (Chapter 14 will explain standard deviation.) In the halves count, each additional count per deck also gives you an extra 0.56% advantage

Halves Win rate: $17 Std dev: $426

Table 68 Comparison of High-Low and Halves Counts Card 2 3 4

5 6 7 8 9 10,J,Q,K

ace

High-Low Halves 0.5 1 1.0 1 1.0 1 1 1.5 1.0 1

o o

0.5 0.0

o

- 0.5

-1 -1

-1.0 -1.0

150


according to Thorp's book, but with a standard deviation of only 0.06%. These numbers come from comparisons of the counting systems with Thorp's card values. The low standard deviation means that the halves count makes almost no errors in estimating advantage. Peter Griffin in The Theory Of Blackjack says the betting correlation of the halves counting system is 0.99 and the playing correlation is 0.58. Simulation shows the halves count results in higher w.in per hour by $1 for the benchmark rules, when compared to the high-low. The sample sizes were increased to the point where that difference is statistically significant. The high-low count overestimates the values of 2 and 9 and underestimates the values of 5 and 7. The halves count is a very close approximation of the true value of each card. As an example, suppose you are watching two tables simultaneously, both tables are single deck, and both dealers are shuffling. One dealer is a bit careless and shows you the bottom card, a 5. The other dealer is even more careless and shows you both the burned card and the bottom card, and they are both 2s. You should immediately make a bet at one table because you have an advantage, but which table? According to Thorp's data, one 5 gone gives you a greater advantage than two 2s gone. The halves count tells you accurately that the table with the 5 gone is better, since it gives you counts of +1.5 and +1. The high-low count gives you the wrong answer in this case, since it gives you counts of +1 and +2. (Griffin claims that a 5 is worth almost exactly the same as two 2s.) An attractive feature of counting halves is that the strategy tables are similar to those of the high-low system. Once you are expert with the high-low system,

HALVES COUNT

151

you can switch to counting halves in easy stages. For example, you can count halves but make your playing strategy decisions by using the high-low tables that you have already memorized; and, if counting halves is too difficult at fIrst you can always switch back to the highlow system. When you feel so confident counting halves that you know you do not want to go back to the high-low system, then learn the halves strategy numbers and forget the high-low strategy numbers. Table 69 is the halves strategy numbers for dealer stands on soft seventeen; it is the counterpart of table 9 (page 46). Table 70 is the halves strategy numbers for dealer hits soft seventeen; it is the counterpart to table 10 (page 48). Insurance Insurance is a good bet if the halves count per deck exceeds +1.7 in a single-deck game, +2.7 in a doubledeck game, +3.2 with four decks, and +3.3 with six decks. Over/Under 13 The over is a good bet at counts per deck of +5 and higher. The under is a good bet at counts per deck of -6 and lower. Appendix B Appendix B contains strategy rules for the halves system. Table B1 is for a single deck when the dealer hits soft seventeen. Table B2 is for four decks when the dealer hits soft seventeen. Table B3 is for a single deck when the dealer stands on soft seventeen. Table B4 is for four decks when the dealer stands on soft seventeen.


152


Table 69 Halves Strategy Numbers, S17 Player's Hand A-A 10-10 9-9 8-8 7-7 6-6 5-5 4-4

spl -8

3-3 ace-9 ace-8 ace-7 -/db ace-7 bIace-6 ace-5 ace-4 ace-3 ace-2

8** 7 6 5

20 19 18 17 16 15 14 13 12

spl

spl -10 spl -5 -3 spl

spl spl

spl spl

spl

9 6

-8 -7 h 0 never split - play as total never split - play as total -2 spl 8* 3 0 3 0 -3 -8 spl

10 7 0

8 5 -1

6 3 -4

5 1 -7

1 -8

1 h h h h

-2 3 6 7 7

-5 -2 0 1 3

-9 -6 -5 -1 0

db db -10 -5 -2

h h h h h

h h h h h

-9 -5 -3 0

-10 -6 -4 -1 2 db -9 0

-10 -7 -4 0 db db -6 1 10 h h

9 10 h h h -9 -6 3 h h h h

8 h h h h -7 -4

1

2-2

hard hard hard hard hard hard hard hard hard 11 10 9

Dealer's Upcard 5 7 8 6 3 2 4 double not allowed after split spl spl spl spl spl -10 -9 8 6 5 5 -4 -4 5 -9 -1 -2 -3

3 db

-8 0 h h h h

8 h h h

-8 -5

-3 0 db -10 -2 5 h h h

-9 -6 -4 -1 db db -4

3 9 h h

9

10

A

-8

-8

-4

-10

spl


5* h h

3 spl h h

h h

h h

h h h h h h h

h h h h h h h

h/1 h h h h h

5 9 h h h -4 -1 h h h h h

0 4 8 h h -4 4 h h h h h

-6 8 9 h h h 1 4 h h h h h

4

8 h h h h

153

HALVES COUNT

Table 69 Continued Player's Hand A-A 10-10

9-9

Dealer's Upcard 7 8 5 6 3 2 4 double allowed after split spl spl spl spl spl -10 -9 5 5 8 6 3 -6 -6 -3 -4 -9 -2

8-8 7-7 6-6 5-5 4-4** 3-3

spl

2-2

-2

-9 -2

h 0

9

10

A

-8

-9

-4

8* h h

3 spl h h

-10

spl spl spl spl spl spl 2 h spl spl spl spl -9 h -4 -6 -8 spl h h never split - play as total of ten h h h 7 1 -2 -5 -5 h -8 -10 spl spl h 4 -4 -6 -8 -10 spl h

spl

h

h h

h h h

KEY Stand. Do not hit. See the -/db row for the number saying whether to stand or double down. db Double down. Hit. h h/Do not double. See the hI- row for the number saying whether to hit or to stand. spl Split. Dealer stands on soft seventeen. s17 number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. (Note that the 1 for ace-7 against ace is hit/stand, and does not say to double down.) Reverse the meaning. Split only if the count per deck * is less than the number in the table. With 4-4, splitting is better than doubling down if ** doubling is allowed after splitting. -/db

154


Table 70 Halves Strategy Numbers, H17

10-10 9-9

Dealer's Upcard 6 3 5 7 8 4 double not allowed after split spl spl spl spl spl -10 -9 8 6 5 4 -3 -1 -4 -6 5 -9 -2

8-8

spl

7-7 6-6 5-5 4-4 3-3 2-2 ace-9 ace-8 ace-7 -/db ace-7 hIace-6 ace-5 ace-4 ace-3 ace-2 hard 20 hard 19 hard 18 hard 17 hard 16

-7 1

Player's Hand A-A

2

8 6 10 7 0 1 h h h

h

spl spl spl -8 -10 spl

-2

-5

3 5 6 7

-2

spl

0 1 3

-9 -6 -5 -2

0

db db db -6 -3

10

A

-8

-9

-5

5* h h

2 -1 h h

h h

h h

h h h h

h h h h h h

-10

spl


h h

h h

h h h h

h

h h h

h h h

h

h h

9

8

0

10

h h

5 9 h

h h

h

-4

-10

hard 14 hard 13 hard 12 11 10 9

8** 7

6 5

spl spl

-3 -5 -8 0 h never split - play as total never split - play as total 8* 3 0 -6 -2 -3 -6 spl 0 2 8 6 5 4 0 4 3 1 -5 -7 db -1

-9 -6 -3 0 3 db -9 1

hard 15

spl spl

9

-7

-8

-4 -1 1 db -9 0

-5 -3 0 db -10

h

8

5

h h

h h h

h h h

h

-2

-9 -7

-4 -1 db db -4 3 9 h h

-9 -6 -3 db db -6 1 9

h h -9 -6 3 h h

h h

h h

h

h h -7 -4

8 h h

h h

h h

-4 -1 h h h h h

3

4

5

10 h h -4 4

8

h

h h h h

h h

0 3 h h h h

h

155

HALVES COUNT

Table 70 Continued Player's Hand A-A 10-10

9-9 8-8 7-7

6-6 5-5

4-4** 3-3

2-2

Dealer's Upcard 9 8 5 6 7 3 4 2 double allowed after split -8 spl spl spl spl spl -10 -9 4 8 6 5 3 -9 -10 -4 -5 -3 -6 -8 spl spl spl spl spl spl spl spl 2 h spl spl spl spl spl spl h h -6 -8 spl h -2 -4 never split - play as total of ten h -7 h h h 6 1 -2 h h -7 -10 spl spl 0 -4 h -4 -6 -8 -10 spl 4 -2

10

A

-9

-5

h h

1 -1 h h

h h h

h h h

8*

KEY

Stand. -I db


db

Double down.

h

Hit. h17 Dealer hits soft seventeen. h/Do not double. See the hI- row for advice on whether to hit or to stand. spl Split. number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. Reverse the meaning. Split only if the count per deck * is less than the number in the table. With 4-4, splitting is better than doubling down if ** doubling is allowed after splitting.

156


CHAPTER 13 DOUBLE EXPOSURE In double exposure, both of the dealer's cards are face up for you to see before you play your hand. This sounds great, but there is another rule that is to your disadvantage: The dealer wins pushes unless you both get naturals. At most casinos with double exposure, naturals win even money instead of three to two but your natural wins even if the dealer also has a natural. Blackjack Secrets has a chapter on double exposure. That book has basic strategy and high-low strategy numbers from -1 to +6. A few paragraphs of this book are word-for-word copies from that chapter of Blackjack Secrets. This book has three things the Blackjack Secrets chapter does not have: high-low strategy numbers for a

DOUBLE EXPOSURE

157

broader range (-10 to +10), strategy numbers for the halves counting system, and simulation results.

Estimating Your Edge Simulation shows a basic-strategy player's approximate expectation for double exposure for four or more decks, dealer stands on soft seventeen, no resplits, and naturals pay even money, is -0.2%. (The negative sign means the casino has the edge.) This is shown in table 71. The rest of table 71 contains numbers that you can add to that -0.2%, depending on how the rules you face vary from those benchmark rules. For example, if you are playing a double-exposure game in which the dealer

Table 71 Player Edge for Dbl Exposure benchmark (both dealer cards face up, pushes lose except natural against natural, dealer stands on soft seventeen, double any first two cards, no resplits) -0.2% to the above add the following: rule change in player edge dealer hits soft seventeen naturals pay 3:2 tied naturals win resplits allowed double after splits no split unlike 10s double ten & eleven only double nine, ten & eleven only

-0.4% 2.3 0.2 0.3 0.2 -0.2 -1.3 -0.9

158


hits soft seventeen, the effect on your expectation is -0.4%, bringing your total expectation to -0.6%. If you get paid though your natural is tied by the dealer's natural, table 71 says you gain 0.2%. If naturals pay 3:2, the value of getting paid on a tied natural is 0.3%. Table 71 shows that allowing double after splits gives the player 0.3%. Of that, 0.1% comes from doubling on less than nine. So if you can double after splits but only on totals of nine or more, you gain 0.2% rather than 0.3%.

Basic Strategy Tables 72-74 contain generic basic strategy for double exposure. Table 72 contains the part of basic strategy for double exposure that applies no matter what the dealer does on soft seventeen. In this table, split strategy is the same whether or not you can double down after splitting. Table 73 is the rest of basic strategy for use if the dealer stands on soft seventeen. Table 74 is the rest of basic strategy for use if the dealer hits soft seventeen. Thus a complete basic strategy for double exposure is table 72 plus either table 73 or table 74.

There are important differences in strategy that depend on whether the dealer hits or stands on soft seventeen. One important difference is if the dealer stands on soft seventeen, then A-6 is played according to the column labeled 17 in table 72. If the dealer hits soft seventeen, there is a separate column for that hand (in table 74). With soft eighteen against small dealer totals, it does not matter much whether you hit or stand if you

159

DOUBLE EXPOSURE

cannot double. Eighteen is not a strong hand when the dealer wins ties. You might be wondering why basic strategy says stand on sixteen against eight at double exposure, but hit at single exposure. Standing on sixteen has equal value at either variety of blackjack - you win if the dealer busts, and you cannot push. Hitting sixteen is more valuable at single exposure because you might end up with a push. The dealer wins ties at double exposure. Seventeen is the worst hand you can get at double exposure - you cannot push, and the only way you can win with seventeen is if the dealer busts.

Counting Cards Each count per deck is worth about 0.7% at double exposure (see page 290). This is in contrast to the 0.5% per count per deck at single-exposure blackjack. Your advantage at double exposure is more volatile than your advantage at single exposure - at higher counts your advantage is higher. Therefore, for equivalent bet sizes you can make more money at double exposure than at single exposure. Counting Cards: High-Low The high-low counting system, used in this book for single-exposure blackjack, is also the best simple counting system for double exposure. This chapter presents

double-exposure strategy numbers for the high-low count. Table 75 provides the strategy numbers that apply no matter what the dealer does on soft seventeen. Table 76 is for use if the dealer stands on soft seventeen. Table 77 is for use if the dealer hits soft seventeen. All the calculations in tables 75-77 are based on six decks. (text continues on page 166)

160


Table 72 Double Exposure Basic, Part 1 Player's Hand

Dealer's Total 7

8

9

10 11 12 13 14 15 16 17 18 19 20

double not allowed after split A-A 10-10 9-9 8-8 7-7 6-6 5-5 4-4 3-3 2-2

soft soft soft soft soft soft soft soft soft hard hard hard hard hard hard hard hard hard hard 11 10 9

8 7 6 5

21 20 19 18 17 16 15 14 13 21 20 19 18 17 16 15 14 13 12

spl spl spl spl

- spl spl spl h h h h db db h h h h h h

-

-

h h h h h h

h

h h h h h h

-

h h h h h

h h h h h

h h h h h h

h h h h h h

h h h h h h

h h h h h h

spl spl spl spl spl spl spI spl spl spl spl db db spl spl spl spl spl spl

-

dbs dbs db db db db db

dbs dbs db db db db db db

spl spl spl spl spl spl db spl spl spl

spl spl spI spI spl spl db spl spl spl





h

h

h

-

spl h h h h h h h

h h h h h h h h

h h h h h h h h h h

h h h h h h

h h h h h h h

h h h h h h h h

h h

h h h

h

h

-

spl spl spl h h spl spl

-

h h h h h

-

-

h

h h

h h

h h h

h h h h

h h

h

h

h

h

h h

h

h

h

h

h h h h h

h h h

db db db db db h h h h h h h h h h h h h h h h

h h

h

h h h

db db db db h h h

db db db db h h h

db db db db db db db



h h h h

h h h

h h h h h h h

h h h h h h h

h

h

h h h

h h h h

h h h

h h h

h

h

h

h h

h h

h

h

161

DOUBLE EXPOSURE

Table 72 Continued Dealer's Total JO JJ J2 J3 J4 15 J6 J7 J8 J9 20 double allowed after split

Player's Hand

7

8

9

A-A 10-10

spl spl spl spl

9-9 8-8 7-7 6-6 5-5 4-4 3-3 2-2

- spl spl spl h h h h db db h h h h h h

h h h h h h

spl spl spl spl spl - spl spl h spl spl h spl spl h db db h spl spl h spl spl h spl spl

h

-

h h h h h h

-



spl spl spl spl spl spl spl spl spl spl

h

h

h

spl h h h h h

h h h h h h h h

-

spl spl spl h h

spl h spl h

KEY db


dbs h spl

Double down; if you cannot double, then stand. IIit. Split.

h h h h h h h h h h

162


Table 73 Double Exposure Basic, S17 Player's Hand

4

5

Dealer's Total 6 A-A A-2 A-3 A-4 A-5

double not allowed after split spl spl spl A-A 10-10 spl spl spl 9-9 8-8 spl spl spl spl spl 7-7 spl spl spl 6-6 5-5 db db db 4-4 h h h spl 3-3 h h spl 2-2 h h soft 21 soft 20 soft 19 dbs dbs dbs soft 18 soft 17 h db db h db db soft 16 h h db soft 15 h h h soft 14 h h h soft 13 hard hard hard hard hard hard hard hard hard hard 11 10 9 8 7 6 5

21 20 19 18 17 16 15 14 13 12 db db h h h h h

db db db h h h h

db db db h h h h

spl

spl

spl

spl

spl

spl

spl

h h h h h h

h h h h h

h db h h h

h db h h h

db h h h

h h h h h h

h h h h h h

h h h h h h

h h h h h h

h h h h h

h h h h h h h h h h

h h db h h h h h h

h db db h h h h h

h db db h h h h h

db db h h h h h

163

DOUBLE EXPOSURE

Table 73 Continued Player's Hand

4

5

Dealer's Total 6 A-A A-2 A-3 A-4 A-5

double allowed after split spl spl

spl

spl spl spl spl db h spl spl


A-A 10-10 9-9 8-8 7-7 6-6 5-5 4-4 3-3 2-2


spl

h h h h h h

spl

spl

spl

spl

spl

spl spl

h

h

h h h h

db h h h

h db h h h

db h h h

spl

KEY db dbs

Stand. Double down; if you cannot double, then hit. Double down; if you cannot double, then stand.

h

IIit.

spl s 17

Split. Dealer stands on soft seventeen.

164


Table 74 Double Exposure Basic, HI7 Player's Hand

4

5

Dealer's Total 6 A-A A-2 A-3 A-4 A-5 A-6

double not allowed after split A-A

10-10 9-9 8-8 7-7 6-6 5-5 4-4

3-3 2-2 soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 soft 14 soft 13 hard 21 hard 20 hard 19 hard 18 hard 17

spl

spl

spl

spl spl spl db

spl spl spl spl db

spl spl spl spl db

h h h

h h h

h

spl spl

dbs dbs dbs db db db db db h h h db h db h h

h

h

spl

spl

h h h h h

h h h h h

h

h

h

h

h h

h

h h h

h

h

spl

spl

spl spl

spl

spl spl spl

db

db

db

h h h

h h

h h

h

h

h h h

h h h

h h

h h h

h

h h h h

h h

h

h h h

h

h h

hard 16

hard hard hard hard 1] 10 9

h h h h h h

15 14 13 12

h

h db

db db h

db db db

db db db

h h

h

h

8 7

h

h

h h

h h h

h h h

h

6 5

h h h h

h

h

h

h h h

h h h h db db h

db db h

db db h

h h

h h h h

h h

h

h h h h

h h

h h h

h

165

DOUBLE EXPOSURE

Table 74 Continued Dealer's Total

Player's Hand

4

double allowed after spl A-A 10-10 spl 9-9 spl 8-8 spl 7-7 spl 6-6 db 5-5 h 4-4 spl 3-3 spl 2-2

5

6

A-A A-2 A-3 A-4 A-5 A-6

split spl

spl


spl spl spl spl db spl spl spl

spl

spl

h

h

h

h h h h

h h h

spl

db h h h

spl

spl

spl

spl spl

spl spl

spl

db h h h

h h h h h h

db h h h

KEY Stand. db


dbs

Double down; if you cannot double, then stand.

h h 17 spl

IIit. Dealer hits soft seventeen. Split.

166


Blackjack Count Analyzer analyzes the many variations of double exposure, and was the tool used to develop the strategy numbers in this book. For decision purposes you need a count per deck. The method for finding it is the same as was discussed in chapter 3. Start with zero after a shuffle. Keep a running count, counting all cards you have seen since the shuffle. Divide this running count by the number of decks you have not seen to get the count per deck. You should stand, hit, or double down only if your count per deck (truncated, and not rounded) is equal to or greater than the number in the table. Insurance. Insurance at double exposure is the same as is explained in chapter 4, and the strategy numbers of table 11 (page 53) are applicable to double exposure too. Generally double exposure uses six decks, which means an insurance strategy number of 3.0. What the numbers mean. You might want to double down on a natural against fourteen, fifteen, or sixteen if the count is high enough. This is true only if naturals pay only even money. Do not double down on a natural if it pays 3:2. Suppose you have 4-4 and have a choice of hitting, splitting, or doubling down. Regardless of the dealer's action on soft seventeen, meaning you are using either table 76 or 77, doubling down on your total of eight is more profitable than splitting 4-4 if the dealer has six or less, or soft seventeen or less. If the dealer has hard twelve to sixteen, then basic strategy says splitting 4-4 is more profitable than doubling down on eight. When the count per deck gets above the numbers shown in table 75, then doubling on eight is more profitable than splitting 4-4.

DOUBLE EXPOSURE

167

Against twelve or thirteen, hit 4-4 if the count per deck is bad enough. The count per deck must be at least -2 to justify splitting 4-4 against twelve, and it must be at least -7 to justify splitting 4-4 against thirteen. With 6-6 against seventeen, split if the count per deck is less than +2 and hit if the count per deck is +2 or more. With 3-3 and 2-2, splitting is always more profitable than doubling down. The numbers for soft twenty against twelve and thirteen are stand/double. If the count per deck is equal to or greater than the number in the table, then double down. With soft eighteen against twelve through sixteen, double if you can. The table numbers for soft eighteen against twelve through sixteen are hit/stand if you cannot double. For example, if you have soft eighteen and the dealer shows thirteen, double if you can. If you cannot double, then hit if the count per deck is less than zero or stand if the count per deck is zero or more. The numbers for soft eighteen against dealer A-A through A-5 are for when to double. When the dealer stands on soft seventeen, you have soft eighteen, and the dealer shows A-A through A-5, if you do not double it does not matter whether you hit or stand. So take a card on a negative count to hasten the shuftle, and stand on a positive count. With 7-7, if you do not split then either hit or stand

according to the strategy number for fourteen. Likewise with 6-6: If you do not split, then play according to the strategy number for twelve.


168


Table 75 High-Low, Dbl Exposure, Part 1 Player's

Dealer's Total JO JJ J2 J3 J4 15 16 17 J8 J9 20 double not a/lowed after split

Hand

7

8

9

A-A 10-10 9-9 8-8 7-7 6-6 5-5 4-4 3-3 2-2

-7

-6

-6

soft soft soft soft soft soft soft soft soft hard hard hard hard hard hard hard hard hard hard

21 20 19 18 17 16 15 14 13

-4

spl spl spl spl spl 2 0 -2 -4 -9 9 spl - -8 spl spl spl spl spl spl h/- h/- h/- spl spl spl spl spl h/- h/- h/- h/- h/- -9 spl spl spl spl hI- h/- hI-hi- hI- - lOs pIs pis pis pI never split - playas total of ten h h h h h -2/5 -7/4 4 4 2 h h h h h -6 -10 spl spl spl h h h h h -6 -10 spl spl spl h

2 -2 h 0 h db h db h -10 h -6 h -4

-

h h h h h

h h h h h h

h h h h h h

h h h h h h

10 6 3 0 -2 -4 -5 -8 -10 0 0 0 db db db db db db db db db db db db -8 db db

1 -9 dbs 0 db db db db db

7

h h h h

7 h h h h h

-9* -8* h spl-l0* h spl h h

h h h

4

-

spl spl -3* spl h 2* h

8

-

h h h h h

h h h h h h

h h h h h h h

-

21 20 19 18 17 -7 16 0 15 5 14 h 13 h 12 h

-6 0 4 h h h

-2 10 2 h h h h

-8 -4 -2 1 7

-8 -10

11

-6

-4

-1

10 9 8 7 6 5 4

-4

-1 h h h h h h

3 h h h

h h h h h h h h

h h h h h h h h

db db db db db 10 db db db db db 8 -7 -10 db db db h 0 -3 -7 -9 db h 5 0 -2 -6 db h 7 2 -1 -4 db h 7 3 0 -4 db h 8 3 0 -3 db h

6 h h h h h

-8 -2 I

h

h h

-6

-

-

h h h h h h

h h h h h h h h

h h h h h h h

h h h h

h h h h h h h h h

h h h h h h h h

h h h h h h h h

h h h h h h h h

h h h h

169

DOUBLE EXPOSURE

Table 75 Continued Player's

Dealer's Total JO Jl 12 J3 14 15 J6 J7 J8 J9 double allowed after split

Hand

7

8

A-A 10-10 9-9

-7

-6

7 spl 8 hi-

spl spl h/hi-

8-8 7-7 6-6 5-5 4-4 3-3 2-2

9

-4

h

spl spl spl spl spl 2 0 -2 -4 -9 -9 spl spl spl spl h/- h/- h/- spl spl spl spl spl h/- h/- h/- -9 spl spl spI spI hi-hi- hi- -lOs pIs pIs pis p I -6

4

7

- spl spl -3* spl h 2* h

8 h h h h

never split - playas total of ten h h h h h -6 spl spl spI spl -9* -8* h 8 h h h h -9 spl spl spl spl spl-IO* h 5 h h h h -9 spl spl spl spl spl h h

20

7 h h h h

h h h h

KEY

Stand. db Double down; if you cannot double, then hit. dbs Double down; if you cannot double, then stand. h Hit. h/db Hit or double; never split. h/Hit or stand. Never split. spl Split. number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. Split if the count per deck is less than the number; * else hit. two numbers see text.

Dbl exp benchmark Win rate: $27 Std dey: $437

170


Table 76 High-Low, Dbl Exposure, 817 Player's Hand

Dealer's Total 4

6

5

A-A A-2 A-3 A-4 A-5

double not allowed after split A-A 10-10 9-9 8-8 7-7 6-6 5-5 4-4 3-3 2-2 soft soft soft soft soft soft soft soft soft hard hard hard hard hard hard hard hard hard hard 11 10 9 8 7 6 5

spl 8 -2 spl 1 0

spl 6 -3 spl -1 -3

spl 6 -3 spl -5 -5

-3

-4

-5

-6 10 2 -1 7 6

9 6 4 7 3 1 h/- h/- h/h/- h/- h/never split - play as total of ten never split - play as total of eight -1 3 0 h h 8 h -1 h h 8 3 0 h

21 20 19 18 17 16 15 14 13 21 20 19 18 17 16 15 14 13 12

8 6 -2 3 5 6 6 6

6 3 -5 -1 0 1 2 3

6 3 -7 -6 -5 -1 0 2

h h h h h h

h h h h h h

9 h h h h h

10 8 4 9 10 10 10 10

-6 -4 -2 0 1 4

-8 -6 -4 -2 0 2

-10 -7 -5 -4 -2 0

-1 2 h h h h h

-2 0 8 h h h h

-4 -1 5 h h h h

-10 -9 -6 -2

-10 -8 -4

-8 -4

-5 -3 -1 0 3 6

db -8 1 9 h h h

db -9 0 6 h h h

db db -3 4 h h h

0 4 h h h h h

-8 9 0 -2 5 4

7 6

9 7 2 7 9 10 9 9

-9 -7 -5 -1 -6 -3 4 10 h h h

171

DOUBLE EXPOSURE


Dealer's Total 4

5

6

A-A A-2 A-3 A-4 A-5

double allowed after split -6 -4 spl spl spl -3 -5 10 6 6 10-10 8 1 8 6 3 -3 9-9 -5 -5 5 2 -2 spl spl spl 0 8-8 -4 8 4 -8 b/- h/-2 7-7 b/- h/8 4 6-6 -7 -3 -5 never split - play as total of ten 5-5 4-4 0 h h h 9 5 1 8 5 -2 h h 0 3-3 -5 -2 h h 8 5 2-2 0 -5

A-A

-8 9

0

-3 2

1 9

3 3

KEY Stand. db Double down; if you cannot double, then hit. h Hit. h/Hit or stand. Never split. spl Split. s17 Dealer stands on soft seventeen. number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number.

172


Table 77 High-Low, Db) Exposure, H17 Player's Hand

4

5


double not allowed after split -6 -5 spl spl spl -4 -6 A-A 8 6 5 10-10 9 -6 9-9 -2 -4 7 1 4 0 spl spl spl h/7 4 8-8 1 -1 -4 h/- h/- 10 7-7 1 6 -6 h/- h/- 10 0 -3 6-6 6 never split - play as total of ten 5-5 never split - play as total of eight 4-4 -1 h h 3 0 h 7 3-3 -2 2-2 3 0 10 h h 7 soft 21 5 soft 20 8 6 9 1 soft 19 5 3 10 7 -9 soft 18 -3 -5 h 9 h 5 -6 h 3 -1 h h 8 soft 17 soft 16 4 0 -5 h 9 h h 5 -2 9 soft 15 1 h h h soft 14 6 2 0 h h h 9 soft 13 6 3 0 h h h 9 hard 21 hard 20 hard 19 hard 18 hard 17 -9 -10 hard 16 -6 -9 -10 -7 hard 15 -4 -6 -5 -8 hard 14 -9 -10 -2 -5 -3 -7 hard 13 -7 -8 0 -3 -4 -1 -3 -4 -7 hard 12 3 0 1 -1 db db db 11 0 -1 -3 -4 db -8 10 -9 4 2 0 -1 9 -3 1 0 h h 8 5 8 9 6 4 h h h h 7 h h h h h h h h 6 h h h h h h h 5 h h h h h h

-8 9 -1 0

-3

4

-9 h h

6 5

h h

9 6 3 6 7 8 8 8

h h h h h

-9 -7 -3 -6 -3 4 10 h h h

3 6 8 h h h 0 2 h h h h h

4

173

DOUBLE EXPOSURE


4

5


double allowed after split spl

A-A 10-10

8

9-9

-3

8-8 7-7 6-6

spl -2

spl 6

spl

-5

-7 spl

-4

-5

-6

6

3

1 2

5

5 9 -8 h/- h/-5 -8 h/- h/-3 never split - play as total

5-5 4-4

4 0 0

3-3 2-2

spl -4 1 -2 -2

-1

-5 -5

h h h

h h h

8 7

-6 9 0 0 4 3

-8 9 -2 -1 2 1

-3 -9 h h

of ten h

7

6

8

5

3

7

4

2

h h h

KEY db

h h17 h/spl number

Stand. Double down; if you cannot double, then hit. Hit. Dealer hits soft seventeen. Hit or stand. Never split. Split. Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number.

174


Counting Cards: Hi-Opt The hi-opt is another simple counting system. In it, 3, 4, 5, 6 count +1, lOis -1, and the· other cards count zero. Earlier editions of Professional Blackjack presented strategy numbers for double exposure for hiopt. That feature has been removed from this book because simulation of it as a simple count (i.e. no second count of aces) gave a win rate of $26, which is lower than that of the high-low. Counting Cards: Halves Simulations show that the halves counting system handily outperforms the high-low at double exposure. Moving up to the halves count for deciding when to bet big means a bigger increase in expected win rate than learning strategy numbers. For decision purposes you need a count per deck. The method for finding it is the same as was discussed in chapter 3. Start with zero after a shuffle. Keep a running count, counting all cards you have seen since the shuft1e. Divide this running count by the number of decks you have not seen to get the count per deck. Insurance for the halves count at double exposure is the same as is explained in chapter 4, and the strategy numbers of table 11 (page 53) are applicable to double exposure too. Generally double exposure uses six decks, which means an insurance strategy number of 3.3. For the halves count, use table 78 with either 79 or 80. Use tables 78 and 79 if the dealer stands on soft seventeen. Use tables 78 and 80 if the dealer hits soft seventeen. The calculations for tables 78-80 are based on six decks. Hit or stand decisions with soft eighteen against four to six or twelve to sixteen are insensitive to the

DOUBLE EXPOSURE

175

halves count. For all of these dealer hands basic strategy is neutral - it does not matter whether you hit or stand whatever the count.

Win Rates at Double Exposure Table 81 contains simulation results for double exposure. The format is the same as for the rest of the win rates in this book. The first row is benchmark win rates, and the rest of the rows are win rates that differ from the benchmark in the identified manner. The benchmark is six decks, five dealt out; ties lose; dealers stand on soft seventeen; player can double any first two cards but not double after splitting; no resplits; naturals pay even money, and tied naturals push. The benchmark rules are used in simulations of the highlow and the halves counts. For simulations of deviations from the benchmarks, results are presented both as raw expected-win rates and as deviations from the appropriate benchmark win rates. Note that varying bets with the count wins almost as fast when you use basic strategy as when you vary your strategy with the count. That means all the strategy numbers of this chapter put together have only a modest effect on your win rate.


176


Table 78 Halves, Dbl Exposure, Part 1 Player's

Dealer's Total 7 8 9 10 11 12 13 14 15 16 17 18 19 20 double not allowed after split

Hand A-A 10-10 9-9 8-8 7-7 6-6 5-5 4-4 3-3 2-2 soft soft soft soft soft soft soft soft soft hard hard hard hard hard hard hard hard hard hard

21 20 19 18 17 16 15 14 13

-7

-6

6

-5

h

spl 2 9 spl -9 sp 1 sp 1 h/- h/- h/- spl h/- h/- h/- h/- h/- -9 h h h h h/- spl

5

never split - playas total of ten h h h h h -8 spl spl 3 2 -9* -6* h h h h h h -7 -9 spl spl spl spl -6* h h h h h h -6 -8 spl spl spl spl h h

h h h

h h h h h

h h h h h h

h h h h h h

h h h h h h

h h h h h h

2 -2 0 db db db -8

8 0

-5 0

db db db -10 -5 -7

5 -2 -7 0 db db db db -10

3

-5 dbs 0 db db db db db

1 -9 dbs 0 db db db db db

10

-

6

-


h h h h h

8

h h h h h h

h h h h h h h

h h h h h h h h

h h

h h h

h h h h

h h h h h

h h h h h

h h h h h

h h h

h h h h h h h h

h h h h h h h h

-

-3 2 7 h h

-6 -2 1

9 h

-8 -4 -1 1 -10 8 -9

11 10 9

-6 -3

-4

-1

-I

6

h

8

h h h

h h

3 h h h h h h

h h h h h h h h

h h h h h h h h

4

spl spl -5 -9 spl spl spl spl spl spl spl spl

h h h h

-

-9 0 5 10 h h

6 5

spl -2 spl spl spl spl

9 h h h h h

21 20 19 18 17 -6 16 0 15 5 14 9 13 h 12 h

7

spl 0 -10 spl spl spl

-

h h

h h h

-

-

-

db db db db db db db db db db -7 -8 db db db 0 -3 -6 -10 db 5 0 -2 -6 db 7 2 0 -4 db 2 0 -4 db 8 8 3 0 -4 db

h h h h h h h 8 h h h h h h

h h

h h h

177

DOUBLE EXPOSURE

Table 78 Continued Dealer's Total 7 8 9 10 11 12 13 14 15 16 17 18 19 20 double allowed after split

Player's Hand A-A 10-10 9-9

8-8 7-7 6-6 5-5 4-4 3-3 2-2

-7

-6

-6

-

7 spl spl spl h/6 h/- h/h h h never split h h h h h h 3 h h

spl spl spl spl spl 2 0 -2 -5 -9 - -10 spl spl spl spl h/- h/- spl spl spl spl spl h/- h/- spl spl spl spl spl h h/- s pis pis pIs pis P1 - playas total of ten h h -8 -10 spl spl spl h h spl spl spl spl spl h h -9 -10 spl spl spl -5

h

8

9 h


h h h h

h

-9* -6* spl -6* spl h

h h h

h h h

5 10

-

6

-

h h h

KEY

db dbs h h/db h/spl number

*

Stand. Double down; if you cannot double, then hit. Double down; if you cannot double, then stand. Hit. Hit or double; never split. Hit or stand. Never split. Sp lit. Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. Split if the count per deck is less than the number; else hit.

Dbl exp halves Win rate: $31 Std dev: $447

Dbl exp hi-opt Win rate: $26 Std dev: $412

178


Table 79 Halves, Double Exposure, S17 Player's Hand

Dealer's Total 4

5

6

double not allowed after split spl spl spl A-A 6 6 7 10-10

9-9

-2

-4

-3

8-8

spl

spl

spl

7-7

1 0

-1 -3

-4

6-6

5-5 4-4 3-3 2-2 soft 21 soft 20 soft 19 soft 18 soft 17 soft 16 soft 15 soft 14 soft 13 hard 21 hard 20 hard 19 hard 18 hard 17

hard 16 hard 15 hard 14 hard 13 hard 12

11 10 9 8 7 6 5

A-A A-2 A-3 A-4 A-5

-4 10 7

-4 6

2

-5

-6

3 0 9 9

10 2 -1 7 7

h/- h/h/- h/never split - play as total of ten never split - play as total of eight 9 3 0 h h h -1 h h 9 3 0 7 -1

7 5 -2 2

4 5 6 6

-7 -5 -3 -10 -7 0 8 h h h

-6

6

6

3 -5 -1 0 1 2 3

3 -7 -6

-4 -1 1 2

-9 -7

-10 -7

-5

-5 db db

db

-9 -1 6 h h h

-3 4 h h h

10 3

1

10 7

h h h h h

h h h h h

h h h h h

-5

-6

-3

-4 -2

-8 -6

-1 0 2 6

0 5 h h h h h

-8 9 0 -2 5 5 8 6

9

8

8

5

2 7

9

10

9

h h h

10 h

10

1

0

-8 -6 -3 -1

4 -1 2 10

2

0

-4 -1

-2 0 7

-4

-6

-1

-3 4

h h h h

h h h h

0

-4 -2

6 h h h h

-9 -6

h h h h

179

DOUBLE EXPOSURE


Player's Hand

4

5

6

double allowed after split spl spl spl 6 6 7

A-A A-2 A-3 A-4 A-5

-4

-4

-5

9 6 h/h/-

5 2

2

A-A 10-10 9-9

-3

-5

-5

8-8

spl

spl

spl

7-7 6-6

-2 -3

-4 -5

-8 -8

5-5 4-4 3-3 2-2

never split - play as total h 5 1 0 h h h 0 -2 -5 h 9 0 -2 -4

h/10

0 7 6

-6 10 1 -2

4 4

-8 9 0

-3 2

of ten h h 6 8 5 7

1 h

4 3

KEY

Stand. db h h/spl s17

Double down; if you cannot double, then hit. Hit. Hit or stand. Never split. Split. Dealer stands on soft seventeen.

number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number.

180


Table 80 Halves, Double Exposure, HI7 Player's Hand

Dealer's Total 4

5

6

A-A A-2 A-3 A-4 A-5 A-6

double not allowed after split -5 -5 -7 A-A spl spl spl -4 10 5 10-10 7 6 1 0 7 3 9-9 -2 -4 -6 3 6 spl spl spl h/1 8-8 7 -1 7-7 1 -4 h/- h/- 9 -6 h/- h/8 0 6 6-6 -3 never split - play as total of ten 5-5 never split - play as total of eight 4-4 10 3 0 -2 h h 8 3-3 -2 9 2 0 h 7 2-2 h soft 21 10 6 5 7 soft 20 5 3 1 9 soft 19 8 8 5 -2 -5 dbs h h soft 18 2 -6 8 h h soft 17 -1 h 3 0 -5 h 9 soft 16 h h 10 -2 h soft 15 4 1 h h 0 h 10 6 2 h h soft 14 6 h h 10 soft 13 3 1 h hard hard hard hard hard hard hard hard hard hard 11 10 9

8 7 6 5

-8 9 -1 0 5

4

spl h h

7 5

h h

9 7

3 6 8 8 9 9

21 20 19 18 17

-9

-10

16

-6

-7

-9

-4 -2 0

-5 -3 -1 1 -1 2

-7 -4 -2 0 -2

-4

0

-1

-6 -3

10

8 h h h h

6 h h h h

h h h h

15 14 13 12

-8 -6

-3 -10 -7 0 8 h h h

-9 -7 -5 db -9 -1

6 h h h

-9 -8 db db -3 4 h h h

3 0 5 h h h h h

h h h h

-3

h h h h h

3 6

-9 -6 -2 -4

-8 -6

-4 4

9 h h h 0 2 h h h h h

181

DOUBLE EXPOSURE


Player's Hand

4

5

6

A-A A-2 A-3 A-4 A-5 A-6

double allowed after split spl 6

spl

-3

-5

spl -2

spl -4

-7 spl

A-A 10-10 9-9

spl 7

8-8 7-7 6-6

-4

-5

6 10

3 5

-8 h/- 10 -9 h/- 10 -3 -5 never split - play as total h 4 1 h -2 0 -6 h h -3 h 0 9 -2 -5

5-5 4-4

3-3 2-2

-5

5 1

2 6 6

-7 10 0 0 4 4

-8 9

-3

-2 -1 2 I

spl h h

h h h

of ten h 7 6

8

7

5

3

4

2

KEY Stand. db

h h/-


Hit. Hit or stand. Never split. hI7 Dealer hits soft seventeen. spl Split. number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number.

182


The average bet sizes for table 81 are $26.47 for the high-low and $26.80 for halves. These are initial bets only, and do not reflect additional amounts wagered on doubles, splits, insurance, etc.

Table 81 Double Exposure Win Rates High-Low Raw Dill Double exposure benchmark 27 BJ pays 3:2 91 64 Resplits allowed 30 3 Double after split 37 10 Player BJ beats dealer BJ 33 6 Dealer hits soft 17 17 -1 0 Double 9, 10, & 11 1 -26 Double 10 & 11 -11 -38 42 cards not used 32 5 62 cards not used 25 - 2 Basic strategy with counting 26 -1 for bet variation no split J-Q 17 -10

Halves Raw Dill 31 98 67 35 4 39 8 37 6 21 -10 4 -27 -8 -39 35 4 26 -5 31 22

0 -9

183

CHAPTER 14 FINER POINTS

Expected Wi n How much money you can expect to make is a function of many things. Your expected win is proportional to your bet size and to the amount of your table time. The more that you bet the more you will win on lucky streaks and the more you will lose on unlucky streaks. Since you must be financially as well as emotionally prepared to handle an unlucky streak, the more that you bet the more you must take with you to the casinos and the more you must have as total capital. I once got a call from a fellow who wanted to take $500 to Reno for a weekend, and win a couple of thousand dollars. He asked which book of mine he should buy to accomplish his goal. I told him sorry, but

184


to win a couple of thousand in a weekend requires bigger bets than can be justified with a $500 bankroll.

Finding Win Rate Per Hour The win rates in the little rectangles sprinkled throughout this book are reasonable estimates of what you can win in an hour, if you are willing to bet up to $100 in the best situations. If you play against the benchmark rules, with the benchmark betting scheme, and are willing to make $100 bets on high counts, you can expect to win $16 an hour. If you want to win at a faster rate, you must: a) Make bigger bets in those situations where you have an edge, or b) Find a game with better rules, or c) Find a game with fewer decks, or d) Find a game with better penetration, or e) Play more hands per hour. Table 82 summarizes the information in most of the rectangles throughout this book. It does not contain double-exposure win rates, and it does not contain win rates for some of the rarer rules. It also contains many win rates that do not appear in rectangles elsewhere in this book. Every number in table 82 was generated by simulation of at least 30 million hands of blackjack. That means the standard error on every number in table 82 is less than $1. In many cases the sample sizes are more than 100 million. Even with samples sizes of 100 million, differences of $1 are too small to be significant. Benchmark rules for one, two, and six decks are each based on samples of more than 600 million hands, and each has a standard error of $0.20. Thus the two-deck win rate of $19 is significantly higher than the six-deck win rate of $16. Both are one deck cut off

185

FINER POINTS

Table 82 Effects on Win Rates One Deck Two Decks Six Decks Raw Dill Raw Dill Raw Difl 48 19 Benchmark 42 13 -6 Dlr hits soft 17 -6 51 3 23 4 Double after split Double 10 & 11 only 38 -10 12 -7 Double 9, 10, & 11 44 -4 16 -3 6 23 Late surrender 54 4 77 29 Early surrender 42 23 Lose all to dlr BJ 15 42 -6 -4 No insurance 41 16 -3 -7 47 No resplits -1 18 -1 78 30 Over/under 13 31 12 96 48 Bustout 42 23 Plr BJ beats dlr BJ 59 1 I 28 9 Dealer's 10 up + ace is ordinary 21 56 8 25 6 6-card win 51 3 22 3 Winning 5-card 21 wins double 50 2 21 2 5-card half win 63 15 33 14 Double on any number of cards, including after splits 61 13 28 9 42 cards not used (21 on single deck) 59 11 27 8 62 cards not used (31 on single deck) 33 -15 10 -9 Hi-opt count 40 -8 13 -6 Halves count 50 2 20 1 Basic strategy with high-low for bet variation 37 -11 14 -5

16 11 20 10 12 21 39 11 14 15 32 30 26

-5 4 -6 -4 5 23 -5 -2 -1 16 14 10

24 18

8 2

17 30

1 14

26

10

20

4

13 12 17

-3 -4 1

12

-4

186


You can use table 82 to find win rates for combinations of things not considered as a group in this book. For example, if you are playing blackjack at a casino that allows double after split and late surrender and also has its dealers hit soft seventeen, simply combine the appropriate win rates. There are several ways to do this, and if you have done the arithmetic correctly all methods should come up with the same answer: $20 per hour. One way is to start with the benchmark $16, add $4 for double after splits, add $5 for late surrender, and subtract $5 for dealers hitting soft seventeen. The average bet sizes for table 82 are $31.48 for one deck, $27.66 for two decks, and $26.54 for six decks. These are initial bets only, and do not reflect additional amounts wagered on doubles, splits, insurance, etc. Risk The discussion of risk should not be separated from the discussion of expected win. If you want to double your expected win you have to bet twice as much, and that means having ups and downs that are twice as large. Those little rectangles sprinkled throughout this book have numbers for standard deviation as well as win rate. Those numbers are all around $400. That $400 describes the ups and downs that are typical of an hour of blackjack play with $100 maximum bets. Actually, the $400 number is conservative because the benchmark describes making your maximum bet at counts per deck of +4 or more, and such high counts do not happen often. If you are more aggressive on raising your bets, and are willing to go to your maximum ($100 in this example) at a count per deck of +2, then the standard deviation of an hour of play is around $500 instead of $400. If you go to $1 00 bets at counts per deck of + 1 or more, then the

187

FINER POINTS

standard deviation applicable to an hour of your play is over $600. If you go to $100 bets at all positive counts, the standard deviation is $750. If you bet $100 on all nonnegative counts, the standard deviation is $800. These numbers are for playing one hand at a time. If you make the same bet on each of two or more hands simultaneously, your risk will be higher yet. The easy way to put it all together is to relate all the numbers to your big bet. Your hourly win rate will be 10% to 50% of your big bet. The standard deviation applicable to an hour of play will be four to eight times your big bet. More on risk later in this chapter. Expected

Value Expected value means the average of all possible outcomes. To find the expected value, find all of the possible outcomes (positive and negative) and the probability of each outcome's occurrence. Multiply each possible outcome by its probability and sum up the products. Your sum is the expected value. For example, suppose you have a natural against an ace and you have seen no other cards in a one-deck game. What is the expected value of insurance in this situation? If the dealer has a lOin the hole, your insurance bet wins double. If the dealer has any other card in the hole, you lose your insurance bet. The probability of the hole card's being a 10 is 15/49, and thus the probability of your winning double the insurance bet is 15/49.

Possible Outcome -1

+2 Sum

Probability Product 34/49 - 34/49 30/49 15/49 1 - 4/49

188


The sum of the possible outcomes times probabilities is -4/49. If you insure all of your naturals, you lose eight cents on every dollar of insurance. (The above calculation is for single deck, but multiple-deck calculations yield the same loss of eight cents per dollar of insurance.) The gambler insures a natural to lock in a certain winner; you should be looking at the -8% expected value. This is a simple example of expected value. For most blackjack decisions, calculating the probabilities is more complicated. The definition of expected value is the same - the sum of: possible outcomes times the probabilities of those outcomes. Thinking in Terms of Expected Win Learn to associate expected value with your bet. If you have a 2% advantage and a $50 bet, think of that as being a $1 expected-win situation. In a single hand you are likely to lose or to win $50; but, over many such hands, you will average $1 in winnings per hand. If the dealer has a 1% advantage and you are betting $50, think of that as giving the dealer 50 cents. Thinking in this manner will eliminate any temptation to bet large when the dealer has the advantage. Whether you won or lost the previous hand does not alter the value of this hand. If the expected value says that you are giving the dealer 50 cents on this hand, then you are giving away 50 cents even if you have just gotten a natural or have won three hands in a row. The same logic applies to other casino games. The dealer's advantage on the pass line at craps is 1.4%; $10 on the pass line is giving the casino owner 14 cents. Thinking in this manner will take all the fun out of gambling and help turn you into an investor.

189

FINER POINTS

Suppose you have a 1.5% advantage on a particular hand of blackjack, and suppose you are betting $50; your expected win on that bet is seventy-five cents. If you play two hands, in total they are worth a dollar and a half. Your bankroll will go up and down in increments of $50 and $100; but, over the long haul, your average win will be about 75 cents per $50 hand in that situation. You must fight one hand at a time, but the object is to win the war; the war is decided by expected values. After you have mastered the strategy, a satisfying way to defme the battle is by use of a target number of hours played rather than a target number of dollars won. To play the target number of hours is to win the battle.

Risk You can win at this game, but you will not win every session. An expected-win rate of $16 per hour does not mean that you will win exactly $16 in any given hour. Far from it. You are as likely to lose $399 in an hour as you are to win $431 in an hour; each is a deviation of one standard deviation from your expected win. Results between those two numbers are what you can expect to see about two-thirds of the time. It will not be rare to see a swing of two standard deviations, e.g. losing $814 in an hour or winning $846 in an hour. That is the risk with no bets over $100, and those big bets coming only on counts per deck of +4 or more. Suppose you play blackjack with the benchmark betting scheme for 40 hours. What is the standard deviation that describes your ups and downs? The answer is easy to find: Simply multiply the one-hour standard deviation by the square root of the number of

190


hours you play. For forty hours, multiplying $415 times the square root of 40 yields $2625 as the standard deviation. For the expected win, multiply the number of hours times the expected win per hour. For 40 hours at an expected win of $16 per hour, the total expected win is $640. An interpretation of these numbers is in 40 hours of blackjack with the benchmark rules, two-thirds of the time your result will be between a loss of $1985 and a win of $3265. Finding Standard Deviation Statisticians use standard deviation and variance to describe the dispersion of possible outcomes around the expected value. One way to find the variance is to subtract the expected value from each possible outcome, square these differences, multiply each square, in tum, by its probability of occurring, and sum. The sum is called the variance, and the square root of the variance is called the standard deviation. Recall the insurance example, in which the insurance bet is lost if the dealer's hole card is other than 10, and double money is won if the hole card is 10. The expected value was found to be -8%. The standard deviation for this example is calculated as follows. (ev is expected value, sq is square, and prob is probability.) possible outcome sq outcome less ev -.92 .85 -1 2.08 4.33 +2 Sum

prob 34/49 15/49 1

product .59 1.32 1.91

Each possible outcome is listed. From each possible outcome, the expected value is subtracted. Those differ-

191

FINER POINTS

Table 83 Area in the Tail of a Normal Distribution

z = outcome less expected value, then divided by the standard deviation Z

.0 .1 .2 .3 .4 .5 .6 .7 .8 .9

shaded area .50 .46 .42 .38 .34 .31 .27 .24 .21 .18

Z

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

shaded area .16 .14 .12 .10 .08 .07 .05 .04 .036 .029

Z

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

shaded area .023 .018 .014 .011 .008 .006 .005 .004 .003 .002

ences are squared. Then they are multiplied by their probabilities. Then those products are summed. The variance is found to be 1.91. The standard deviation is the square root of the variance, or 1.38. Over a large number of decisions of this type, say, more than

192


a couple of hundred, the distribution of possible outcomes approaches the normal. Normal Curve The normal distribution is useful. It is symmetric and has a standard shape. You can tell from the normal distribution as shown in table 83 how likely a given outcome is if you have an expected value and a standard deviation. The expected value plus or minus one standard deviation contains the actual outcome about 68% of the time. The actual outcome will be worse than one standard deviation below the expected value 16% of the time, and the actual outcome will be better than one standard deviation above the expected value 16% of the time. Here is a sample use of the normal curve. What is your chance of being down by more than $1000 after four hours of play for the benchmark rules and betting scheme? Your expected win is $16 per hour, or $64 for four hours. To be down by $1000 instead of up by $64 is to be $1064 below your expected result. The standard deviation applicable to your play is $415 for an hour of play, and it goes up with the square root of the number of hours. So the standard deviation for four hours of play is $830. Dividing the amount away from the expected result by the standard deviation gives the number of standard deviations you are away from the expected result: $1064/$830, or 1.3. Table 83 says that the area in the tail beyond 1.3 standard deviations is .10, meaning there is a 10% chance that you will be behind by $1000 or more after four hours of benchmark play. Living With Risk When deciding how much to bet, pay special attention to the standard deviation because it quantifies the

FINER POINTS

193

risk you will be taking. You must be able to sleep with the risk. You will have streaks where you lose 20, 30, or 40 big bets over the course of several days of playing blackjack. If you do not ride easy with such unwholesome and sticky burrs, you will have some grisly dreams. I once lost $5,500 betting $100 per hand on the final day of a four-day outing, which cut down the trip's win from $12,000 to $6,500. This was back in the 1970s, when $6000 bought a nice car. If you cannot stand the thought of losing several thousand dollars, do not bet $100 per hand. Start small. You must build confidence in the system and in your ability. Getting a few dollars ahead will help. You must get accustomed to thinking of money as a means of keeping score rather than for what it will buy. Think about buying later. Compartmentalize such thinking away from professional matters. To professional blackjack players, betting $100 per hand and ending up $6,500 ahead after having been up by $12,000 means being 65 bets up after having been up by 120 bets. If your only impression upon reading these dollar figures was that I was two automobiles ahead after three days and on the final day dropped one automobile to finish one automobile ahead, you are not yet ready to bet big yourself. If you thought of the nwnber of bets instead of the purchasing power and you have the required capital, then go ahead and bet big.

Bankroll Advice First, decide how much money you are willing to risk at blackjack. By that I mean the amount that if you lost it all you would give up the game. That is your total stake for the purposes of this discussion. The bet size that is mathematically optimal is discussed later in this chapter. Personally, I prefer to bet

194


more conservatively than can be justified by mathematics. This paragraph presents the approach to bet sizing I use myself. I prefer to relate the biggest bet I am willing to make to my total stake. When I want to play one hand at a time, I hold my big bet to 1/150 or less of my total stake. When I want to play two hands at a time, I hold each of two simultaneous big bets to 1/200 or less of my total stake. So when I was willing to risk $30,000, I was willing to bet up to $200 on one hand, or $150 on each of two hands. For a short trip to casino country, a day or a weekend, a sufficient bankroll is 50 times whatever you select as your big bet. If you want to bet up to two hands of $100, then take $5000 with you (or be able to write checks of up to $5000.) For an extended blackjack trip, I recommend taking at least 100 times whatever you select as your big bet. Doubling your bet size doubles your expected win, but also doubles risk and doubles capital requirements. You can only increase your expected win without increasing capital requirements by playing more hours. Playing two weekends a month will yield twice the expected win (and twice the expenses) on the same trip cash and the same total stake. If you want to make a lot of money playing blackjack, you are going to have to play a lot and bet a lot. You must be able to lose a lot of money. You do not want to lose, but you will lose frequently. You will win frequently too, and over the long haul, you are certain to win more than you lose, but the long haul may be a long time in coming. You simply must be secure enough financially and emotionally to handle losses that will occasionally amount to the

FINER POINTS

195

trip-cash numbers and may, in a very bad streak, add up to your total stake. The smaller your big bet in relation to your bankroll, the lower the probability that you will be wiped out. Your blackjack bankroll behaves much like a portfolio of stocks. A mathematician would describe it as following a random walk with an upward drift. Your possible outcomes from investing in blackjack are normally distributed. Investors in the stock market do not have any assurance that their outcomes are normally distributed; their possible outcomes appear to come from a distribution that has a greater chance of extremely good or extremely poor outcomes.

Nadir To fmd out how far in the hole $50 top bets could put you in a few days of blackjack, I have run four simulations. This is a rewrite of an article I wrote for the March 1990 issue of Blackjack Forum. The rules used are benchmark plus double after splits and no resplits. Other sets of rules would yield similar results. There was one player at the table, who used the high-low with strategy numbers from -1 to +6. Whenever the count went negative, the player switched tables. On counts per deck of zero to Wlder +1, the bet was $5. On counts per deck of +1 to under +2, the bet was $20. On counts per deck of +2 or more, the bet was $50. Different bet schemes would yield different results. The study was done for both six and eight decks. With six decks, the shuflle came with a deck and a half left. With eight decks, the shuffie came with two decks left. 1000 hands were played, and a record was kept of the lowest point the accumulated winnings (or losings)

196


reached during those hands. Then the bankroll was brought back to its initial position, another 1000 hands were played, and another lowest point was registered. This procedure was repeated until 20,000 lowest points had been tallied. This was done for both six and eight decks. Then the whole procedure was repeated, but with 4000 hands played before the lowest point is recorded and the bankroll was reset. Thus a total of 200 million hands of blackjack were played for this study. Results. The results are shown in table 84. Each row shows the probability of being down by a given amount or more at the lowest point of the session. For example, after playing 1000 hands at eight decks, 19.7% of the time your lowest point would have been a loss of $1000 or more. The data in table 84 are simulation results; each true frequency could be slightly higher or lower. The standard errors on the numbers in table 84 range from almost zero to 0.35%. You can find the individual standard errors by taking the square root of p(l OO-p)/20000, where p is a number from table 84. For example, the standard error on the 19.7% is the square root of (19.7)(80.3)/20000, which is 0.3%. Having an edge means you will be ahead for certain if you play long enough, but it does not guarantee a win in the short run. 4000 hands is a short run. Comparison of the four columns of table 84 shows that there is more risk in playing 4000 hands than in playing 1000. No surprise there. A surprising result is that risk seems to depend almost entirely on the number of hands played, and almost not at all on whether the play was against six or eight decks. There was slightly more risk playing against six decks than playing against eight. That is because there

FINER POINTS

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Table 84 Pro babilities of Low Points Most dollars ever behind never behind $1 or more $100 or more $200 or more $300 or more $400 or more $500 or more $600 or more $700 or more $800 or more $900 or more $1000 or more $1100 or more $1200 or more $1300 or more $1400 or more $1500 or more $1600 or more $1700 or more $1800 or more $1900 or more $2000 or more $2100 or more $2200 or more $2300 or more $2400 or more $2500 or more $2600 or more $2700 or more $2800 or more $2900 or more $3000 or more $3500 or more $4000 or more $4500 or more $5000 or more $5500 or more $6000 or more

1000 hnds 1000 hnds 4000 hnds 4000 hnds 6 decks 8 decks 6 decks 8 decks 1.0% 1.0% 0.6% 0.5% 99.0 99.0 99.4 99.5 87.2 86.5 91.9 92.0 77.5 76.0 85.6 85.6 68.2 66.4 80.0 79.6 59.6 57.5 74.5 74.2 51.7 49.5 69.0 69.0 44.4 42.1 64.2 63.7 37.6 35.3 59.5 59.0 31.8 29.3 54.9 54.0 26.6 24.3 50.4 49.5 21.9 19.7 46.4 45.3 17.9 15.9 42.6 41.6 14.8 12.7 39.0 37.8 12.0 10.2 35.5 34.4 9.6 7.9 32.2 31.2 7.6 6.2 29.3 28.3 6.0 4.8 26.4 25.5 4.7 3.6 23.7 23.0 3.7 2.7 21.6 20.7 2.9 2.0 19.5 18.6 2.3 1.4 17.7 16.6 1.7 1.0 16.0 14.7 1.2 0.8 14.5 13.0 0.9 0.5 13.0 11.7 0.7 0.4 11.7 10.3 0.4 0.3 10.4 9.0 0.3 0.2 9.3 8.0 0.3 0.1 8.0 7.0 0.2 0.1 7.0 6.2 0.1 0.1 6.0 5.5 0.1 0.0 5.2 4.7 0.0 0.0 2.6 2.2 0.0 0.0 1.3 1.0 0.0 0.0 0.6 0.4 0.0 0.0 0.3 0.2 0.0 0.0 0.1 0.1 0.0 0.0 0.0 0.0

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were more opportunities to make big bets against six decks. Big losses come from being unlucky on big bets. The expected win is greater for six decks, one and a half cut off than for eight decks, two cut off; that greater profit opportunity greatly outweighs the increased risk. Likewise, though there is more risk the longer you play, there is more expected win the more hands you play. Remember that table 84 is for bets ranging from $5 to $50. Higher bets are accompanied by more risk, but also of course by more expected win. Table 84 is based on bets of $5, $20, and $50 at specific counts per deck; more conservative betting would be accompanied by less risk. Table 84 is based on one person at the table, playing one hand at a time. More people at the table probably would not change the risk much, but playing more than one hand at a time definitely would result in more risk. Trip cash requirements. Table 84 says that for 1000 hands, a bankroll of 50 big bets has a risk of ruin of about half a percent. For a play of 4000 hands, a starting bankroll of 90 big bets is required to reduce the risk of ruin to less than half a percent. Those numbers assume no cutting back in bet size in response to losses. For pessimists. The odds are excellent that sometime in the near future your playing bankroll will be less than it is at this instant. For example, table 84 shows that there is a 99% chance that at some point in the next 1000 hands your bankroll will be less than it is right now. How far is your bankroll likely to fall if your trip is sort of average? From table 84 you can find the low points that occur with about a 50-50 chance for the $5 to $50 bet scheme of this study. For 1000 hands, a dip of $500 is about average. For 4000 hands, a dip of $900 is

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about average. Remember that these are worst-point intermediate results, and not end-of-trip numbers. Other statistics. The expected wins (simulation results) for the four columns of table 84 are $226, $167, $904, and $668. The standard deviations are $1028, $960, $2056, and $1920. If you want, you can compare the nadirs with the final distribution of results as described by the normal curve. Comments On Risk Peter Giles says: The late Ken Uston was once quoted in the Review Journal as saying, "It's really tough to make a living at blackjack. The fluctuations will really wipe out the average guy. If I had to play by myself (instead of on a team), I probably wouldn't be in it now." You can quote me as saying the same. I have chosen to not use an optimal betting system. I would rather start with enough units to have a very low risk factor and keep the bets the same. I am tired of having to make up losses by winning more bets than I lost. I know the reverse is also true, but it still has a negative effect on players on a team. I know one team that lost $45,000 in two days, lowered its bets, and ground it out for weeks. The players would show up every day in a bad mood, and it may well have affected their games. Cutting their bet sizes meant they had to win twice as many bets to get back to even in dollars. I am still trying to determine how many units one is safe with. What is generally recommended in books is, in my opinion, too risky. This is one area in which it is hard to trust mathematics.

Donald Schlesinger says: I have the utmost respect for both Ken Uston and Peter Giles, but their pessimistic outlook on the individual card counter's plight is, in my opinion, too harsh.

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Fluctuations are inherent in the game. Precise and careful money management protects the average guy from the wipeout that Kenny describes. The game can be beaten and fluctuations can be overcome. Your readers should take heart, keep the faith, and practice, practice, practice.

Marvin French says: I am back from Reno after a satisfying trip. I started by winning, kept on winning, and ended on an all-time high. Your advice to stay emotionally detached from win/loss results is excellent, but difficult for us humans to follow. I enjoy the game much more when my bankroll goes steadily upward during the trip rather than ending high but with huge swings getting there.

Chance of Reaching All-Time High Unfortunately, most of the time you seem to be losing. According to mathematician Peter Griffin, at most 1.6% of the time your present hand puts you at a newall-time high. The other 98.4% or more of your playing time, your current bankroll is lower than it was at some time in the past. This can be tough on your nerves. If you compare your present position with your all-time high, the feeling that you are losing will haunt you during 98.4% of your playing time. One way to avoid getting depressed is to forget about your all-time high. Chances of moving up are always greater than chances of moving down, but downward movements will continue to occur. Your trend is still up. You must hit a new high sometime if you keep playing well.

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Avoid Going Broke: Use Proportional Betting By now you have probably won several hundred dollars betting small and are wondering whether you should increase your bet size. Perhaps a proportional betting scheme has occurred to you; that is, you are considering betting a fixed proportion of your capital on each hand. Some proportions are better than others. The proportion that is best is called the optimal proportion, and when multiplied by your total stake is called your optimal bet size. In blackjack, the outcome on one round is virtually independent of the outcome of the next round. (The Gwynn and Seri paper presented at the Fourth Gambling Conference found a negative correlation too slight to be of any importance.) This independence is a desirable property in that it simplifies calculation of an optimal bet size. If you make a series of optimal bets, you will never go broke and you will become wealthy as quickly as possible. (In theory. In reality, overhead expense and table minimum leave ruin a serious possibility.) Any other bet size involves a longer expected time to become wealthy and may even include a chance of going broke.

Optimal Bet Size Assuming that you start playing blackjack with a given amount of capital, make no addition to your total stake except for blackjack winnings, and make no subtractions from your total stake except for blackjack losses, your optimal bet size is a proportion of your total

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stake equal to your advantage divided by the variance of possible outcomes. Your advantage changes from round to round. At most casinos the dealer has an advantage over you on the first round after a shuflle. Therefore, you would like to bet as small as possible on the first round. If the count stays zero or goes negative, the dealer still has an advantage over you and if you stay at that table you would like to bet as small as possible. If the count goes positive you may have an advantage. For every unit increase in the count per deck, your advantage increases 0.5% (at standard blackjack; at double exposure, the number is 0.7%). To estimate your advantage, take half of the count per deck and subtract the dealer's initial edge. Example: A) You are playing in a game where the dealer has 0.5% advantage over you on the first round. Suppose the count per deck is now +5. Your advantage is about 2% (-0.5 + 5xO.5). Variance of possible outcomes depends on the particular rules you are facing, as is shown in table 85. Being able to double after splitting means higher variance (bigger ups and downs). Having double down restricted to only ten or eleven means lower variance. Risk also depends on the number of simultaneous hands you are playing. The measure of how likely two hands are to win 'together or lose together is covariance; it also is shown in table 85. (A covariance of zero means the two hands are independent. If you could sit between two blackjack tables and play one hand on each, the covariance between those two hands would be zero. A covariance of 1.00 means two hands win and lose in lock step with each other.)

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Table 85 Variance and Covariance for Blackjack

Variance Covariance

Benchmark 1.28 0.47

Double 10 & 11 only 1.20 0.43

Double After Split 1.32 0.48

The numbers in table 85 for variance and covariance are simulation results generated by Blackjack Count Analyzer. If you play more than one hand at the same time, your optimal bet size changes because two hands at the same table are not independent. Optimal bet size can be found with the aid of table 86. To find your optimal bet size, multiply your total stake times your percent advantage times the appropriate amount from table 86. For example, with $10,000 in total stake, benchmark rules, and a 1% advantage, your optimal bet size is $78 on one hand, or $57 each on two hands, or $45 each on three hands. If the count goes up enough so that your advantage is 2%, your optimal bet size will be $156 on one hand, or $114 on each of two hands, etc. The numbers in table 86 are based on the variance of one hand and covariance between two simultaneous hands. Letting v be variance, c be covariance, and n being number of simultaneous hands, the numbers in table 86 are 1/(v+(n-l)c). This formula is based on the

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Table 86

Optimal Bet as a Proportion of Your Advantage Simultaneous Hands 1 2

3 4 5 6 7

Benchmark 0.78 0.57 0.45 0.37 0.32 0.28 0.25

Double 10 & 11 Only

0.83 0.61 0.49 0.40 0.34 0.30 0.26

Double After Split 0.76 0.56 0.44 0.36 0.31 0.27 0.24

trade-off of risk and return popularly known as the Kelly criterion. If you want to remember round numbers for sizing your bets, perusal of table 86 suggests 80-60-45. That is, for one hand bet 80% of your advantage times your total stake. For two simultaneous hands, bet 60% of your edge times your total stake on each. For three simultaneous hands, bet 45% of your edge times your total stake on each. Note that playing two hands instead of one should mean betting only 50% more money. If consideration of risk and return suggests betting say $200 on one hand, then if you want to play two hands you ought to bet $150 on each for $300 total. If you want to go to three hands, you ought to bet around $112 on each for $336 total. So the total money you can justify betting on one round does not increase much if you want to play three or more simultaneous hands. The reason is the covariance - the

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results of all your simultaneous hands are tied to the dealer's hand. Suppose you play two simultaneous hands. A natural for the dealer means you are likely to lose both of your hands, and if the dealer starts with a small card up and busts you likely will win both of your hands. Your advantage varies from round to round, and varying your bet size as your advantage varies is theoretically desirable. As a practical matter, however, varying your bet size with your advantage will not make you a millionaire; it will get you barred from play. To avoid that cruel fate, you must blend in with the gamblers. How much you should bet is determined by more than just what is mathematically optimal. If you are not emotionally prepared to accept the risk associated with optimal bets, then bet less. If you bet less than the optimal amount, you will never go broke, but you will not get rich as fast as you might. Know yourself - check your emotional readiness to lose before you set up a betting plan. You must be cold and calculating. Steely blue eyes will do. You must have no emotional attachment to that money you throw on the table. It must mean little to you. If you lose it, so what; it is only money. If you cannot afford to lose it, you cannot afford to bet it. In no way is short-run profit guaranteed. You cannot plan on winning the next hand. Lady Luck has a heart of stone. You

will be way ahead if you play long enough, but only if. The actual play is a hand at a time and you must be emotionally able to handle the losses that will occur regularly. Bet an amount you can live without. You must be able to withstand a one-day loss of twenty times whatever is your big bet, because such a loss will occur occasionally.

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Some bet sizes are more convenient than others. For example, suppose your optimal bet size is $9.16. Now $9.16 is an impossible bet size for blackjack. Even $9 is an awkward bet size. Go ahead and bet $10. Rounding off does not hurt. The important thing in optimal betting is to be sure to reduce your bet when your capital shrinks.

Overbetting Perhaps your attitude toward risk is that you are willing to wager more than the optimal amount on each hand. For example, maybe your capital is $1,000 but you are willing to use $25 as your big bet. If you bet more than the optimal bet size, you are overbetting. Overbetting will cause you to win more when you are lucky and to lose more when you are unlucky. To avoid going broke, you would have to cut your bet size after a series of losses. The disadvantage of overbetting is that the extra win when you are lucky is outweighed by the accelerated bet cuts dictated by the bigger losses when you are unlucky. You can overbet for a while and get away with it, but the person who bets the optimal amount consistently will make more money over the long haul than the person who overbets consistently.

Overbetting can be financially fatal. It can be proved mathematically that an overbettor who consistently bets twice the optimal amount will break even over the long haul. The overbettor who bets more than twice the optimal amount will go down instead of up in the long run. If you continually overbet you will go broke even though you have an advantage. Do not overbet.

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If you do not understand why overbetting is bad, deal 100 hands to yourself and pretend bet, say, onefourth of your capital on each round. You must recalculate your capital after each round. Your capital goes down fast, even if you win more hands than you lose.

Stopping Rules A cousin problem of overbetting is what to do when you are losing and running out of pocket cash. Always keep enough money to split a pair or to double down. For example, if you are down to your last $200 and the pack is very profitable, the most you should bet is $100, or $50 for each of two hands. Save the other $100 in case you must double down or split a pair. Try to carry enough cash so that you do not get into this position. Never make a bet that you would be unwilling or unable to match if you should have to split a pair or double down. You only have an advantage over the dealer if you are willing and able to follow the recommendations of the system. Although a time comes during a losing streak when you should stop, namely, when you run out of pocket cash, no rule for stopping governs a winning streak. The probability of your winning the next hand from an honest dealer is the same whether you have been losing or winning. Quitting offers no advantage when you are a certain number of dollars ahead. True, it is disappointing to make only $6,500 on a trip when you were $12,000 ahead at one time, and you may tend to blame yourself for being greedy; you tell yourself that you should have quit with the $12,000. Cheer up - you did the right thing in continuing to play. No optimal stopping rules exist. You never know whether you are right now at a high point or whether you will continue to win in the

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hours ahead. If you are playing correctly, the chances of your winning another thousand dollars always exceed the chances of your losing another thousand dollars. Never become apprehensive, and never become satisfied. If you think you are being cheated, then of course stop betting. Your chance of being cheated in a legal casino in Nevada or Atlantic City is close to zero.

The Optimal Number of Simultaneous Hands Optimal bet sizes, including zero when you do not have an edge, are based on mathematics and computer simulations. These are the amounts you would like to bet if you could bet any amounts you want. But of course you cannot bet anything you want - your bet size is subject to constraints. There are minimum and maximum bets. Only certain bet sizes are accepted. If casino employees are scrutinizing you to evaluate your ability, you might impose additional constraints on yourself. On negative counts: If you cannot bet zero, then bet the minimum you can get away with. If you play a spot, that spot gets cards. If you leave that spot blank, the dealer does not give it any cards. Playing one more hand thus uses more cards, and may mean fewer rounds before the shuffle. For example, if enough cards remain for three more head-up rounds of one hand each, you will get only two rounds if you play two hands each round. Which makes more money, three single hands or two double hands? The answer depends on how much you are betting per hand. It also assumes that the dealer will not have to shuffle until at least one subsequent hand. (If this is the last round before the shuffle no matter how

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many hands you play, then optimally if you have an edge you should spread your money over as many hands as possible.)

Alone and High Limits If you are alone at a table, if you have an advantage, if your optimal bet size for one hand is less than the table maximum, and if another round will be dealt after this one, then it does not matter much whether you bet one hand or two, but do not go to three hands. To go to two hands of optimal bets is to bet about 150% as much per round as to play one hand per round. And to add a second hand is to use about 150% as many cards. You will get 2/3 as many rounds before the dealer shuffles, but you will be betting 150% as much per round, so two hands at a time has the same profit potential as one hand per round. Adding a third hand and making optimal-sized bets would add 12% to your total bets but would use 33% more cards, and thus result in lower overall profit potential. If you are the only customer at a table and the dealer has the advantage, your optimal action if you stay is to play one hand at the minimum. To play two minimum bets at a time would be to bet twice as much on each of the two-thirds as many rounds before the shuffle, for 33% more unprofitable action. Therefore, if you are the only customer at the table, it makes sense to play one hand at a time whether or not you have the advantage. This advice holds if your optimal one-hand bet size is less than the table maximum.

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Alone and Low Limit If you are the only customer at the table and your optimal single-hand bet is more than the table maximum, then your optimal total bet on two simultaneous hands will be more than 150% of the table limit. In this case, you will be better off with two hands per round. You will get two-thirds as many rounds before the shuffle, but you will be betting more than 150% as much per round. One Other Player and High Limit If there are other players at the table with you, your adding an extra hand adds a smaller proportion to the total cards used on one round. You would bet 50% more with two optimally-sized bets instead of one, but with one other player you would use only 33% more cards per round. This means you are better off playing two hands if there is another customer at your table and you have an edge. Do not go to three hands of your own; your third hand would use 25% more cards but your total bets (with optimal bet sizes) would increase by only 12% per round. More Other Players With two or more other hands played besides your own, you are better off with two hands yourself when you have an edge. However, increasing to three hands of

optimal-sized bets will not win you more money no matter how many other players are in action at your table.

Another Argument for Fewer Hands Your optimal bet size changes as your advantage changes, and you would like to estimate your advantage as closely as possible. Playing fewer hands means estimating your advantage more often between shuffles,

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hence more precisely. For example, suppose you are playing two hands at once, and suppose your first hand is 10-10. Your second hand can be thought of as coming from a pack that is poorer by two 10s than the pack you used to estimate your advantage to select your bet size. Had you known in advance that those two lOs would not be available to your second hand and ,to the dealer, you would have bet less on your second hand. The existence of this precision factor suggests being hesitant about playing another spot when a few cards could cause big changes in the count per deck. For example, if one deck is used, you might stick with one spot if you are alone or there is one other player, and switch to two spots if there are two or more other players.

Summary Generally, if you are alone with the dealer you are better off playing one hand, and if there are other players you are better off playing two hands when you have an edge.

All Decisions Are Approximations Every counting system involves approximations. Whatever the high-low system tells you is your advantage is only an estimate of your true advantage. Your true advantage might be slightly more or less than what any count indicates. When the high-low system recommends a certain playing-strategy option, that option is probably (but not always) the best way to play that particular hand. Some more complicated counting systems are capable of making slightly better approximations than the

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high-low system, but only if they are used absolutely correctly. Many users of more complicated counting systems switch back to the high-low after discovering that increased complexity leads to increased errors in casino play, and these errors more than offset any gain from using the more complex count. Even a computer makes approximations. All counting systems, even computers, assume that the next card to be dealt is equally likely to be any of the unseen cards. A computer can make better decisions than you can because it can keep track of how many cards of each rank remain unseen. A computer is capable of manipulating these data to make the best decisions possible given that the unseen cards can be in any order, but such calculations are time-consuming. For much faster but only slightly poorer decisions, a computer that keeps track of how many of each rank remain unseen will still use approximations to arrive at its recommendations. Recently some brilliant people have taught computers to keep track of the cards in the order used, called shuffle tracking. These computers can do better than simply assume the next card is equally likely to be any one of those yet unseen. The simulation results reported in this book are not the best a computer can do. What I asked my computers to do is play blackjack using the counting systems and the strategy numbers presented in this book. Thus the win rates presented in this book are attainable by a human. Though a computer could make decisions based on the exact mix of cards remaining rather than on a counting system and strategy numbers, that has not been done for any of the simulation results reported in this book.

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Which Counting System A major decision a card counter faces is which counting system to use. Here is a letter from a reader that expresses the quandary: I hope you can help me. I am writing to you for some advice. I have been playing blackjack for about a year now and I have been counting cards for most of that time. I love the game and plan to play semiprofessionally, but here is my problem. What is the best count method to use? When starting to play last year, I read Humble and Cooper's book, The World's Greatest Blackjack Book, and I used the Hi-Opt I count. Next I read the late Lawrence Revere's Playing Blackjack as a Business and started using his Hi-Lo because it was not an aceneutral count. I played blackjack in Las Vegas using both counts on different visits and won money using both. A few months ago, I read Bryce Carlson's Blackjack for Blood, and of course he says his basic and advanced Omega counts are best. And then, to make matters more confusing, I read your book, Blackjack Secrets, and you use the Hi-Lo count, but you vary the strategy according to the count and the dealer's hitting or standing on soft seventeen. I know that there are different levels of counts, but which is the best? I consider myself of at least average intelligence and I do not mind putting in the hours of study that it takes to become a professional blackjack player. I just do not want to waste any more time on a blackjack system that is not going to help me be the best player possible. So, there is my dilemma. If you could help me, it would be much appreciated.

My answer to this reader is: It does not matter much which counting system you use, as long as you are using

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something that compares high cards to low cards. The high-low is the best of the simple systems. If you want to use a more powerful system, the halves count wins faster than the high-low, but is also more difficult. Much more important is what kind of a game you can get, specifically how much you can bet on high counts. I suggest that instead of trying to learn a more complicated counting system or memorizing a wider range of numbers than -1 to +6, you would be better off thinking about what kind of an act would allow you to get away with bigger bets when your count finds an edge. Analyze what kind of image you project (or fix yourself up to project the image of your choosing) and then select an act that is consistent with the image you project. You can have multiple acts, each with its own look (clothing, hair, personality, et cetera). Getting away with card counting in a casino is a big and important topic, and it has its own book: Blackjack Secrets. How should aces be counted? According to Peter Griffm, the importance of each card for betting purposes is close to the importance of each card in detection of appropriate deviations from basic strategy, except for aces. The high-low and halves systems presented in this book count aces in the high-card group. This is better for betting purposes, partly because naturals pay three to two. Some other systems, hi-opt for example, count ace as zero in the primary count. For most playing-strategy decisions, more accuracy is achieved if aces are counted as zero. However, for hi-opt to approach the high-low in win rate, a side count of aces must be incorporated for betting purposes. Keeping a side count means keeping

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two running counts, which of course is much more difficult than keeping just one running count. If you use only one count (Le. no side count of aces), should aces be counted in the high group or counted as zero? My answer is to include aces in the high group, with the lOs. The reason is that estimating your edge for bet sizing is more important than knowing when to vary from basic strategy. The simulation results reported in this book provide ample evidence. My simulations of simple counting systems (one count only, and no side count) show that the high-low outperforms the hi-opt whether naturals pay even money or 3:2. The high-low yields a higher win rate than the hi-opt for any number of decks. The sample sizes are large enough that the differences are statistically significant. Halves does better than the high-low, $1 per hour better for the benchmark rules, but is also more difficult to use. Whether you think the improvement in win rate justifies the increase in difficulty is a personal matter. Ever since I developed it in the early 1970s, I have used the halves count with no side counts. For most of my playing career I used strategy numbers from -1 to +6.

How Many Numbers To Learn The text of this book presents strategy numbers from -10 to +10. The appendix presents strategy numbers from -20 to +20. I personally use strategy numbers from -1 to +6. What should you use? Part of the reason I memorize such a limited range of strategy numbers is during the days I supported myself playing blackjack, I traveled the world to find the best games, and I played against quite a variety of rules.

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I learned the strategy numbers appropriate to whatever rules I was facing at the time. Often on a plane flight I would learn the strategy numbers appropriate to the casinos of my destination. For example, sometimes I combined Korea with Macao; I would learn the appropriate Korea numbers on the plane to Seoul, and then replace them in my brain with the Macao numbers on the flight from Seoul to Hong Kong. Another reason for learning a small range of numbers is you are fooling yourself if you think you really need a large range of numbers. Simulations show little if any increased win rate for memorizing strategy numbers less than -1 and greater than +6. As for negative strategy numbers: You ought not to be playing against negative counts, and if you do have to play against them you ought not to be betting much, so why worry about deviating from basic strategy? As for high positive strategy numbers: They occur rarely in an honest game. If you do happen to get a high-low count per deck of say + lOin a shoe game, you ought to be asking yourself whether you are getting an honest game rather than rejoicing in a rare opportunity to use high strategy numbers. My own experience is that the only times I have gotten a count per deck of +10 or more on each of the first two shoes I played, I later found out I was being cheated. I am including strategy numbers from -10 to +10 in this book because people want to buy a wide range of strategy numbers. I can generate them, and I can publish them for you, but that does not mean I recommend that you use all of them. Wm rates for the range of numbers -1 to +6 are only $1 per hour less than the win rates for the benchmark range of -10 to +10.

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The high-low system involves a simple counting rule and a relatively small amount of data to be memorized; yet it gets the money. You can estimate your advantage more accurately, but only by using a more complicated counting system. You can learn a more accurate playing strategy, but only at the cost of memorizing more strategy numbers.

Side Count of Aces The 1981 and earlier editions of this book contained strategy numbers for high-low adjusted for aces and halves adjusted for aces. Using those numbers required keeping a side count of aces. That material is not included in this edition due to its lack of importance in increasing your win rate.

Two-Card Combinations The tables in the text of this book base playing strategy on total points in your hand and not on which particular cards make up your hand. Appendixes A and B contains information on strategy for particular twocard combinations. The additional dollars per hour you can win by knowing two-card strategy numbers is essentially zero, particularly with multiple decks. The most important two-card combination is 7-7 against 10 for a single deck. With the high-low system, for instance, you should stand with 7-7 against 10 with one deck if the count per deck exceeds -1, whereas the

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first part of table Al (page 255) says that you should always hit fourteen against 10. You might examine the two-card combinations to get an understanding of how sensitive certain decision rules are to the removal of key cards. For example, thirteen against 2 is sensitive to 7s and 8s. The greater proportion of 7s and 8 that are left in the deck, the more likely you should be to hit thirteen against 2. The greater proportion of 7s and 8s that you have already seen, the more likely you should be to stand on thirteen against 2.

Resolving Toss-ups Here is a tip for decisions when whether or not to take another card is a toss-up. If the count is negative, take the card; if the count is positive, do not take the card. The reason for this advice is: The more cards that remain to start the next round, the more likely it is that there will be a next round.

Quick and Accurate You will never learn to play perfectly like a computer. The computer will spot correct deviations from basic strategy that you would never catch. For example, a computer can spot situations in which splitting 3-3

against lOis the correct decision; all that is necessary is a big excess of 8s and 9s in the remainder of the pack. The systems in this book say do not split 3-3 against 10, because a count of 8s and 9s is useful for few other decisions. You can never playas well as a computer, just almost as well.

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To win in casinos you must make decisions quickly and accurately. It is not advisable to use a counting strategy in a casino if you cannot play quickly and accurately with it. No strategy modification will yield big gains over the high-low system. If you want to play many hours in a day, you cannot use a strategy modification that you find tiring. Keeping two counts is more tiring that keeping a single count, and keeping three counts is considerably more tiring. You can modify your system from day to day or hour to hour, since all of the decision tables are similar. The use of one set of decision tables with a different counting system or a different number of decks results in few incorrect decisions. I like to use a simple counting system so that I can play long hours, talk with other people at the table, keep an eye on the pit boss, and not get exhausted. When I fIrst started playing blackjack, I used Thorp's 10-count, which involves keeping two running counts. I made money with it, but it was a lot of work. Then I switched to the high-low, and not only did I make money faster, but the effort of counting seemed to be reduced by 90%. Now I use the halves count, which is only a bit more work than the high-low and (for me) worth the extra effort.

Thorp's Complete Point Count You may have compared my high-low strategy numbers with the complete point count system of Ed Thorp in the second edition of Beat the Dealer and wondered why the differences exist. For a valid comparison, cut Thorp's numbers in half because his high-low index is approximately equal to count per two decks. Thorp's rules assume one deck, with the dealer standing on soft

220


seventeen, and no restrictions on doubling down. Thus, you might compare Thorp's numbers with table A3 (page 263). The numbers are not identical, and the main reason is the way Thorp treated 7s, 8s, and 9s. Suppose that Thorp wanted to create a deck that had a count of +5; he would do so by removing five small cards from a full deck. The remaining 47 cards would thus be rich in aces and lOs, but would also be rich in 7s, 8s, and 9s. This procedure distorted some frequently-used decision rules. A person who is using Thorp's rules would tend to hit twelve against 2 too often and would stand on sixteen against 7 too often.

With Other People at the Table A reader says: The count a twenty-one player uses represents the composition of cards not seen, which many times, when not playing head-on, is different from the composition of the cards remaining to be dealt. Since count systems, as I understand, are developed on the basis of head-on play, will not the playing efficiency of any system be significantly affected when playing with other people? If so, should a player attempt to guess the unseen cards in the other players' hands and then adjust the count for that one hand? This is an area no one seems to discuss; therefore I hope you will answer my questions.

The playing efficiency of your count system is not affected by playing with other people at the table. Playing strategy does not depend on how many players there are at the table. The main effect of having more players is slower play; fewer hands per hour means a lower hourly win rate. Yes, you should try to guess the

FINER POINTS

221

unseen cards in other players' hands and use that information in the play of your hand. Penetration is important. Look for games with good penetration. And develop an act that allows you to bet more money when you have an edge than when the casino has an edge.

222


CHAPTER 15 SHUFFLES One way to get an edge is to follow individual cards or groups of cards through the shuffle. Most casino shuffles are so complicated that attempting this is a waste of time. Shuffles are standardized within a casino; whatever shuffle you witness on one table will likely be repeated on other tables that use the same number of decks. Occasionally you find a casino in which the dealers use an incomplete shuffie. That is, some shuftles involve breaking the pack into half-deck-size groups and shuffling those groups two at a time, but otherwise not intermixing the groups. If you count cards, you might be able to have a count for each of several different groups of cards, follow those groups through the shuffie, and cut an excess of aces and lOs to the top.

SHUFFLES

223

New Decks Shuffled, But Not Together Something to watch for if you start a game at a table with unshuffled cards: See if the dealer mixes all the decks together. Sometimes each deck will be shuftled individually, and then the decks will be slapped together (but not shuffled together) and given to you for a cut. Bill Gaem popped into Pop's Oasis in Jean, Nevada. The dealer was standing at the table. In front of him were spread four decks. They obviously were used cards, but they had been sorted and arranged to resemble new decks. Then a customer came. The dealer shuffled each deck individually, but the individual decks were not shuffled with each other. Thus the shoe contained four single decks, stacked one on top of the other. Another method of shuffling new cards, observed at a five-deck game in the Riviera in Las Vegas, is intermixing two decks, then intermixing two more, and lastly shuffling one deck by itself. If you see a dealer shuffle like that, you can watch where the cut card is placed to see whether you are playing against one of the double decks or a single deck. To a basic-strategy player, it makes little difference whether a shoe contains single decks stacked one on top of the other or intermixed decks except that the casino's edge is about half a percent smaller with the single decks. But to a card counter the difference is major. A running count of +4 in a single-deck game calls for a big bet, whereas a running count of +4 early in a four-deck game is still in the ho-hum range.

224


Single-Deck Shuffle Study This is a study of the most common shuffle used at single-deck blackjack in Las Vegas casinos. The sample consists of ten independent shuffles, each of which starts from the same arrangement of cards. Sampling Methodology Dave Douglas, who was a blackjack dealer in Las Vegas casinos for five years, did the shuflling. I asked him to do typical casino shuffles - neither overly careful nor overly sloppy. The shuflling was done on a standard blackjack table at the home of Peter Giles. Casino cards were used. The resultant shuflles are thus representative of what you routinely encounter in singledeck games in casinos. The cards were initially arranged in the order of new decks: by suit, with the ace of hearts on top. Douglas performed ten shuffles, each of which began with the same ordering of the cards, and each of which included the same steps. The shuffle used is rift1e-riffle-strip-riflle. A variety of shuftles are actually used in casinos, and this one is common. It is also about the minimum you find in casinos. Many dealers riffle more than three times or strip more than once. Only rarely will you encounter a dealer who riffles fewer than three times or does not strip. A riffle is breaking the deck into two approximately equal parts and interleaving one into the other. Dealers also call this a "shuffle" but I am using "riffle" to avoid confusion; I am using the word "shuffle" to mean the sum total of the actions to reorder the cards. In this study, the "shuffle" is riffle, riftle, strip, riflle.

SHUFFLES

225

A strip is breaking the deck into six to ten parts and inverting the order of the parts. The dealer holds the deck with one hand and pulls groups of cards off the top with the other hand. Most shuflles also include turning half of the deck around during one of the riffles, but turning the cards end for end does not change their order. Data The shuftle data are presented two ways. Table 87 lists the cards by suit and point value. Examples: DA is diamond ace, CT is club 10, H7 is heart 7. Table 88 has exactly the same data except that each card has been replaced by its ranking in the deck before the shuffle. The first column of both tables is labeled "0." It contains the order of the cards before each of the ten shuffles. Each of the columns labeled 1 to lOis the column 0 deck after a riffle-riffle-strip-riffle sequence. Analysis of Results You might think that the same dealer doing the same shuffle on identical decks would rearrange the cards approximately the same way every time. Comparing columns 1-10 with each other shows that this is certainly not the case. Each shuffle results in a different final arrangement of cards. You might be wondering if any particular card would be more likely to end up in a predictable spot. A test of this is to see how often two shuffles put the same card in the same location. The probability of a card in one shuffle being matched by the same card in the same location in another shuffle is 1/52. There are 52 locations total, and 45 pairs of comparisons possible per location. Multiplying the above three numbers together yields 45, which is the expected total number of times a card in

226


Table 87 Raw Data From Ten Shuffles 0

1

HA

OA CT H7

H2 H3 H4 H5 H6 H7 H8 H9 HT HJ HQ HK CA C2 C3 C4

C5 C6 C7 C8 C9 CT CJ CQ CK OK DQ OJ OT 09 DB

07 06 05 D4 03

02 OA SK SQ SJ ST S9

sa

S7 S6 55 S4 S3

S2 SA

S6

C5 08 CK HT 55

C6 07 S4

2 C4

H7 HJ S7 C9 C5 03 OQ CT OJ 02

sa

OK H3 SK C7 OQ

H6 OA S9 HQ C3

D6

SA

HJ

sa

C4

HQ S7

so

H2 OT SJ ST HK CA 02 SA C2 CJ H8 H9 S9 CQ C3 H4 D9 HA S3

OJ C8 05 H5 S2 D4

H6 C9 03

H4 CJ SJ CQ H5 CK ST C8

C2 HT D4

OK S2 SK 08 H9 SQ S3 CA

07 HA C7 H2 HK H3 DB

05 S6 S5

H8 C6 OT 09 S4

3

4

5

6

7

8

9

10

H7

07 S7 C6 C2 OK

OK

OT C9 H3

C2 OA 08 C3 H8

CK

C4

OJ H9 SA

S6

C6 C7 OT H4 C3 55

05 09 D4 S4

H2 C8 HQ HK OK ST C9 S9

CT DQ CA 03

sa

OJ H8 CQ H9 HT SK 02 HJ C5 CK S2 SQ SJ

DB

CJ HA SA 08 C4 S3

OA H5 07 H6 C2 S7 H3

S6

C7 DQ C3

C8 H2 05 C4

H5 OJ

D4 S5 H6

OT CJ 03 HK SA

sa

CA H4 CQ C5 CK

S6

sa

DQ C5 CK C2 S5

C8 SQ S4

C9 OJ D4

03 CA 02 S3

OA H9 SK H5 S2 D6

HQ SA

H9

05 OT HK HT 09 HA 08 H2 87 H3

D9

H8

06

OA S4

SK H3 S3

SQ 08 S2 HT SJ

HJ ST S9 C9 HQ CT 02 HA

H7 H8

sa

OA H9 C2 CT S7 CJ 09 CQ

OJ H2 02

H9 C5 SK 07 H5 06 H6 H7 02 C6 OJ C9 CA CT S7

S9

D4

DB

CK S6

C3 H4 SA

C7 CA HA H8 HT SQ C8 HJ C5 SJ D4

H4 C6 HJ C7 07 H6

HQ DK S2 DQ C6 ST 03 HK H7 SK

SJ

C3 CT 5T C4

CJ CQ 59 H7

C4

OT CJ 09 HA CQ S9

H4 S3

HK S2 OK SA

sa

S6

S2 H4 HJ DQ SK OJ HQ C7 SQ H2 SA

C8 HK OT CA 09 SJ C9 C2 H6 HA 55

H9 03 OK 02 S4

HT S3

OA DB

H5 H3

03

C4

C7

C3

OQ H2 SJ

05 C8

ST 05 CT C5 S9

S5

HQ H3

H8 CJ D4

C4

S6 55

07 06 H5 S4

H6

S3

05

CK S4

ST HT

so HJ

sa

C6

CQ S7 08 H7 07

OK 02 H8 H9 OA HQ CJ 07 SK S3

CQ S2 HK DB

SQ CA SA SJ

H5

sa

CT H6 HJ S7 S6

06 H7

sa

H4

D4

C7 S7 OT S9

03 H3 C6

HT SJ

C3 C4

HJ ST C5 02 DQ CA C2 H7 HK

co

55

SQ H2 H8

D4

OK

05 OQ H4 C5 C6 OJ

HT

03 OT HA C7 H2 S4

H3 C8 09 C9 C2 ST CK C3 S9

S4

S3

OA 05 HQ 82

SK HA 55 D6

CK CJ 09 C8 C9 H5 S6

H6 CT 08 07

227

SHUFFLES

Table 88 Card Positions of Ten Shuffles 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

31 32 33 34 35

36 37 38 39 40

41 42 43 44 45 46 47 48

49 50 51 52

1 39 23 7 47 18 32 26 10 48 19 33 49 27

3 40 20

28 34

11 45 17 12 46 41 2 30 42 43 13 14

38 52 15 24 8 9 44 25 16 4 31 1 50 29

21 35 5 51 36

6 22 37

2

17 7 11 46

3 7 47 19 20

22

30

18 37 28 23

4 16 48 35

14 33 1 20 2 13 3 34

31 36 49 2 21 12 13 27 43 22 44 23 28 14 37 45 29 8 25 9 10 40 38 11 18 26 51 41 42 34 24 1 52 32 17

35

50

47

39 5 33 6 15 46 3

29

38 45 6 39 44 12 16 52 4 24 42 25 5 26 43 21 15 10 36 27 51 40 32 9 41 50

48

8 19 30

31 49

4 33 46 19 15 27 47 20 28 16 21 2

5

6

27 47 45 28 18 26 15

30

48

48

21 41 49 22 29 36 37 14 38

6

50

30

39 9 40 5 51 34 12 52 35

35

17 5 29 36

24 37 13 52 45 14 4 25 18 26 34 39 49 40 3

30

13 10 31 1 32 2

50

46

9 31 41 32 51 10 42 11 43 44 22 12 23

3 8 4 19 11 20 33 6 42 16 23

38

1 7 8

43

17 24 25 44 7

22 3 45 39 9 15 23 46 24 31 25 32 26 47 16 4 52 29

2 38 44 20 14 1 8 10 41 21 11 18 42 36 12 27 51 28 19 43 37 13 7 40 48

33 17 34 5 49 6 50 35

7 15 39 32 16 8 17 9 18 40 33 5 34 6 7 38 19 29

22 14 23 46 36 30

24 31 1 25 44 4 50 13 51 27 52 45 37 20 28 2 42 35 21 12 3 47 48

26 49 43 10 41 11

8

9

10

26

17 27 38 8 9

29

47 51 4 11 28 40 29

12 20 41 2 52 21 13 30

14 31 42 22 15 6 1 48

9 37 27 38 49 10 50 39 34

5 3 17 16 43 35

23 18 44 8 24 36 45 19 25 46 32 7 33

39

12 24 33 40 50

25 51 13 32 41 14 52 42 5 45 23 6 11 46 47 34 7 48 35

28 36 4 18 19 29 10 37 30 1 20 2 49 3 21 31 22 15 43 26 16 44

9 52 45 4 36 20 46 30

44 37 3 19 10 42 16 17 11 43 18 38 28 14 15 7 13 25 41 2 8 49 27 50 39 35 12 51 40 1 48

34 26 24 31 21 22

5 47 6 23 32 33

228


one shuffled deck is in the same location in another shuffled deck if the cards are completely random after the shuftle. An example is C4 is the top card in shuftles 2 and 9. The actual total of such pairs in the ten shuftles is 48. This can be treated as a Poisson distribution, meaning the variance is approximately equal to the expected number of pair~, or 45; hence the standard error on the actual number of pairs is the square root of 45, or 6.7. The actual number of pairs is close to the expected number. Therefore knowledge of the bottom (or top) card before the shuffle is useless information after the shuffle (assuming the absence of eyesight keen enough to follow the card through the shuffle). Does the above paragraph mean the shuffle provides a random reordering of the cards? Not necessarily. A random reordering means that every card has an equally likely chance of appearing at any position after the shuftle, which seems to be the case, but it also means that every arrangement of cards is equally likely. Here is a test of whether some arrangements of cards are more likely than other arrangements. Look at the pairs of cards that are initially adjacent, and analyze their relative positions after the shuffle. For example, before the shuffie CJ (club jack) follows CT (club 10). After shuffle #1, CT is the second card and CJ is the thirty-fourth card; the number of cards after CT and before CJ is 31. Treat the shuftled decks as if the top card follows the last card; this will happen when the deck is cut. Example: H7 initially follows H6. After shuffle #1, H6 is the fiftieth card and H7 is the third card. After the cut there will be four cards after H6 and before H7. Table 89 lists the final distribution of cards that were adjacent before the shuflle. "Gap" is the number of cards between initially adjacent cards. A gap of zero

229

SHUFFLES

Table 89 Post-Shuffle Gaps Between Initially Adjacent Cards Gap Freq o 23 1 57 2 40 3 43 4 35 5 23 6 18 7 6 8 9 9 6 10 3 11 1 12 5 13 6 14 6 15 1 16 2

Gap Freq 17 5 18 3 19 4 20 9 21 3 22 3 23 6 24 1 25 5 26 8 27 7 28 6 29 7 30 13 31 8 32 10 33 19

Gap Freq 34 11 35 11 36 9 37 10 38 8 39 7 40 8 41 13 42 10 43 6 44 3 45 6 46 4 47 0 48 2 49 0 50 1

means the cards are still adjacent and in the same order. A gap of 50 means the cards are adjacent after the shuffle, but the order is reversed; the only time this happened in the sample is in shuffle #8: C4 ended up directly ahead of C3. If the order of the cards were truly random after the shuffles, the theoretical distribution of the gap between initially adjacent cards would be uniform. That is, a random shuffle would find the 51 0 data points

230


spread over the 51 possible gaps at the rate of about 10 per gap. The sample shows an excess of small gaps between initially adjacent cards. The chi-square value is 646.4 with 50 degrees of freedom, which corresponds to a zstatistic of 26. Therefore you can say beyond doubt that the distribution of cards after the shuffle is not random. Knowledge of the exact order of the cards before the shuffle has value in predicting the very next card to be dealt because small gaps between initially adjacent cards occur more often than could be due to chance. No large clumps of cards were preserved through any of the shuffles. There was not a single instance of three cards staying next to each other. In a completely random shuffle the probability that three cards remain together (or are separated and then reunited) is once per fifty decks, or 0.2 per ten decks; this study got zero, which is not significantly different from expectation. In a completely random shuffle the probability of three initially-adjacent cards finishing up in that order with no more than one card separating them is 0.8 per ten decks; this study got one, which is not significantly different from expectation.

Practical Considerations I receive many letters discussing shuffling, and most of the comments can be put into two categories. Some people are concerned that in some diabolic way the standard casino shuffle works to their disadvantage. There is no evidence in this study to support such a notion. Other people think that the standard casino shuffle is so poor that it can be exploited by players who have knowledge about the order of the cards before the shuffle. In this study, 47% of the time cards that were

231

SHUFFLES

adjacent before the shuffle were in the same order and separated by six or fewer cards after the shufile. Perhaps this could be exploited by a player to gain an advantage. For example, if you memorize the card atop an ace before the shuffle, when that card appears there is a 47% chance its ace will follow within six cards (assuming a single deck).

Speed of Blackjack Single-Deck

Blackjack

This is an empirical study of how fast single-deck blackjack is dealt in casinos at Reno, Nevada. A stopwatch was used to measure the actual time taken to shuffle the cards and to deal to various numbers of players.

Data I timed a total of 42 shuffies and 167 rounds of play for a variety of dealers at a variety of casinos. The results are shown in table 90.

Shuffle Time The time recorded for the shuffle is the total elapsed time from the instant the dealer broke the deck to the instant the first card was dealt on the next round. Shuffle time thus includes time for cutting the deck and burning a card. Shuffle time also includes time for anything else that interrupted the dealer during the shuffle; the most frequent such interruption was for making change. One dealer blew her nose during the shuffle, so her nose-blowing time is included. Without any interruptions, the mean shuffle time would be just under 20 seconds. With interruptions included, the mean time for the 42 shuflles I observed is 23 seconds.

232


Table 90 Average Times for Shuffling and Dealing Single-Deck Blackjack In Reno Number of Players shuffle 1

2 3 4 5 6 7

Sample Size (Rounds) 42 46 21

Mean Time (Seconds) 23 17

23

23 32

19 29 21 8

33 45 50 59

Dealing Time Dealing time was from the instant the first card was dealt to the instant the first card was dealt on the following round or to the instant the dealer broke the deck for a shuffie. There were sometimes short delays, primarily for a player to buy chips; the clock was always running dwing these delays. The players' first two cards were dealt face down. The players could double down only on ten or eleven. All samples were taken at casinos that offered insurance. If a dealer ran out of cards before finishing a round, that sample was aborted. Dealing time depends on the speed of the dealer and the speed of the players. Individual hands also differ; e.g. if the dealer has a natural, the hand is over quickly. This study was done back when dealers checked hole cards under lOs.

SHUFFLES

233

Mean dealing time increased approximately linearly with the number of players. The average round of single-deck blackjack in Reno took about (7n + 10) seconds, where n is the number of player hands. It appears that the average dealer's hand took 10 seconds, or longer than the average player's hand, which was 7 seconds, but this interpretation is incorrect. You cannot separate the time for the dealer's hand from the time for the first player's hand. The mean combined time for the dealer and the first player was 17 seconds. Each additional player took 7 seconds on average to play a hand. However, you cannot say that the first player averaged 7 seconds to play a hand. The reason is second and later players were studying their hands while the dealer was waiting for the fITst player to make a decision. The first player did not have this extra time. Estimates of Rounds per Hour Using the empirical estimates for Reno singledeck games of 23 seconds per shuflle, 17 seconds for the dealer and the first player, and 7 seconds for each additional player, you can estimate the number of rounds per hour for various numbers of players and various numbers of rounds between shuffles. These are only averages of course; the actual number of rounds per hour you will get in a casino will vary somewhat from table 91. In particular, if you find a fast dealer and play fast yourself, you can get considerably more rounds per hour in a head-on game. Discussion of Rounds Per Hour A player of average speed with a dealer of average speed in a head-on game with seven rounds between shufiles gets about 177 rounds per hour. When another player joins the game there will probably be about five

234


Table 91

Average Rounds Per Hour for Single-Deck Blackjack in Reno Player Hands 1

1 2 3 4 5 6

7

Rounds Between Shuffles 234 5 6

7

8

90 126 146 158 167 173 177 181 77 101 114 121 126 129 67 85 93 98 59 73 79 82 53 64 68 48 57 44 51

rounds between shuffles, and the speed will be cut to about 126 rounds per hour. When a third player joins the game, there will be at most four rounds between shuflles; the players will be getting about 98 rounds per hour. When the table fills up, rounds per hour will be cut in half again. Besides slowing play down dramatically, increasing the number of players at a table has another harmful effect: It means fewer rounds between shuflles, and that means playing a larger percentage of the hands with freshly-shuffled cards. The 51 hands per hour at a full table are split - half directly after shuffles and half where you get to see some cards before making a bet. At the head-on game, the 177 hands per hour include about 27 right after shuftles and 150 where you get to see some cards before making a bet. If you disregard the first hand after each shuffle, one average hour at a head-on game yields as many hands as six hours at a full table. If you play quickly, the ratio might be higher than six to one.

235

SHUFFLES

The implication of course is that you are much better off with a dealer to yourself than at a full table.

Six-Deck Blackjack This is an empirical study of how fast six-deck blackjack is dealt in casinos at Atlantic City, New Jersey. A stopwatch was used to measure the actual time taken to shuffle the cards and to deal to various numbers of players. Data I timed a total of 13 shuflles and 261 rounds of play for a variety of dealers at a variety of casinos. The results are shown in table 92.

Table 92 Average Times for Shuffling and Dealing Six-Deck Blackjack in Atlantic City Number of Players

Sample Size (Rounds)

shuffle 1 2 3 4 5 6 7

13 59 53 31 28 38 21 31

Mean Time (Seconds) 100 12 20 27 35 38 42 48

236


Shuffle Time The time recorded for the shuffle was the total elapsed time from the instant the dealer reached for the used cards to the instant the first card was dealt on the next round. Shuffie time thus includes time for cutting the pack and burning a card. Shuflle time also includes time for anything else that interrupted the dealer during the shuflle; the most frequent such interruption was for making change. Shuffles of new cards were not included; they are much slower; each changing of the cards took about ten minutes. Different casinos used different shuffles, but they all took about the same amount of time. The mean time for the 13 shuffles I timed is 100 seconds. Dealing Time Dealing time is from the instant the frrst card was dealt to the instant the first card was dealt on the following round or to the instant the dealer reached for the used cards to shuffle them. There were sometimes short delays, primarily for a player to buy chips; the clock was always running during these delays. The players' cards were dealt face up; the players never touched the cards. Dealing time seemed to depend primarily on the speed of the dealer. Some dealers were very fast; some were capable of 500 hands an hour for head-on play. Mean dealing time increased approximately linearly with the number of players. The average round of blackjack in Atlantic City took about (7n + 5) seconds, where n is the number of player hands. This is about 5 seconds per round faster than single-deck blackjack in Reno; the difference seemed to be due to the fact that the Atlantic City game was dealt face up.

237

SHUFFLES

You cannot separate the time for the dealer's hand from the time for the first player's hand. The mean combined time for the dealer and the first player was 12 seconds. The actual number might be slightly less than 7 seconds, but my sample size is too small to be more precise.

Rounds Per Hour Using the empirical estimates for Atlantic city sixdeck games of 100 seconds per shufile, 12 seconds for the dealer and the first player, and 7 seconds or each additional player, you can estimate the number of rounds per hour for various number of players. Table 93 assumes the dealer cuts off about 100 cards; dealers who cut off fewer cards will deal more hands per hour. These are only averages of course; the actual number of rounds per hour you will get in a casino will differ somewhat from table 93.

Table 93 Average Rounds Per Hour for Six-Deck Blackjack in Atlantic City Player Hands

1 2

3 4 5 6 7

Rounds Per Hour

248 158 116

91 76 64 56

238


Eight-Deck Blackjack I timed 21 shuffies of eight decks at various Atlantic City casinos. The average time was 2:22 for reshuffies of used cards. Shuffles of new cards take about five minutes.

Atlantic City Blackjack A reader adds: What prompts me to write is your articles on speed of play in Reno and Atlantic City casinos. I am developing my own rules of thumb for evaluating win rates per hour at blackjack games. I have noticed that each player takes visibly longer to play a hand at face-down games compared to face up, mainly because in face-up play the dealer anticipates the player's actions. So I was surprised that both of your articles said seven seconds for each additional hand; I would have expected the face-up Atlantic City game to be faster than the face-down Reno game. I ran off least squares linear curve fits on your data, weighting for sample size. For Reno I got (6.8n + 10.0), which closely agrees with your (7n + 10). However, for Atlantic City, the result is (6.0n + 7.6), which is quite different from your (7n + 5).

You are correct in that the actual play of the hands is faster at face-up games than at face-down games. This factor tends to make Atlantic City faster than Reno. There is another factor, however, that works in the opposite direction. The other factor is the time it takes the dealers to do other things, such as make change. These other things take longer in Atlantic City than in Reno. You can bet cash in Nevada, but in Atlantic City you must convert your cash into chips - even for doubling down and buying insurance.

239

SHUFFLES

The sample sizes in table 92 are too small for much precision, but I suspect the underlying relationship is not linear. Nonlinearity is caused by new customers buying chips to join in the game. This effect is much stronger in Atlantic City than in Reno. Games with two to five players seem to be more vulnerable to being slowed down by new customers buying chips. Full tables, of course, cannot have new customers buying in unless a player leaves. Perhaps full tables really average closer to 60 rounds per hour instead of the 56 listed in table 93.

Search For Streakiness Random

Shuffle

Several blackjack authors state that the distribution of outcomes of blackjack games is more "streaky" than one would expect from strictly random outcomes. They say that wins come in bunches that can be predicted by an observant player. They say that some dealers are "cold," meaning easier to beat than a dealer who is "hot." Charles Einstein in his Basic Blackjack Betting advocates "rhythm betting" to take advantage of streakiness. Eddie Olsen developed and Jerry Patterson sold TARGET, a system that in part tries to select tables where the cards are biased in favor of the player and where the dealer is busting more frequently than usual. These are not the only authors who claim you can profit by predicting streakiness. Our brains try to see patterns in things that are random. Common sense seems to support the notion that there are runs of good and bad luck. This is one area where our brains can play tricks on us. What seems to be common sense sometimes is nonsense.

240


None of the authors who say you can make money on streakiness has provided evidence to support the existence of streakiness. Therefore, I have done this study to measure streakiness and the difference between hot and cold dealers to estimate how much money you can make by betting more when you are winning and looking for cold dealers, and how much you can save by avoiding hot dealers. Study This study uses benchmark plus double after split and no resplits, eight decks shuftled, six dealt out, and one player using basic strategy. (There is nothing special about this set of rules; it just happened to be common in Atlantic City at the time the study was done.) The computer keeps track of whether the player wins, loses, or pushes. It also keeps track of the dealer's final hand. It hits the dealer's hand until it gets to seventeen or more even if the player busts or gets a natural, so as to see whether the dealer is hot. Results are kept in sets of threes. The sample size is 20 million. Results - Win, Lose, or Push Table 94 shows the relative frequencies with which wins, losses, and pushes occur as a function of how the player has fared on the previous two hands. For example, the first row is what happens on the next hand after you have won two hands in a row. The overall averages are .432 chance of winning, .478 chance of losing, and .089 chance of pushing, which do not quite sum to 1.000 due to rounding. Though your chance of winning is smaller than your chance of losing, your average win when you win is greater than your average loss when you lose. You win three to two on your naturals, and you win your doubled bets more frequently than you lose them.

241

SHUFFLES

Table 94 Outcomes Versus Results of Previous Two Hands previous two hands winJwin winJlose winJPush loseJwin lose,lose lose,push push,win pushJlose pushJPush

outcome of this hand win lose push .432 .478 .089 .432 .478 .089 .432 .478 .089 .433 .478 .089 .432 .479 .089 .433 .478 .089 .433 .478 .089 .432 .478 .089 .432 .479 .089

The chi-square statistic for table 94 is 6 with 16 degrees of freedom. This means there is no relationship whatsoever between what happens on this hand and whether you have won, lost, or pushed on the previous two hands. Trying to bet more when you are winning or less when you are losing is an exercise in futility; it will not win any money. Streaks occur, but they cannot be predicted. Results - Dealer's hand Table 96 shows the relative frequencies with which dealers get particular totals as a function of what they have gotten on the previous two hands. For example, the first row shows what happens on the hand after the dealer has gotten two consecutive naturals. 21 means all twenty-ones except naturals. The overall averages for

242


dealer totals are shown in table 95. (The total is slightly different from 1.000 due to rounding.) The chi-square statistic for table 96 is 473 with 288 degrees of freedom. This translates to a z-statistic of 7, which means the dealer's outcome is not independent of what the dealer has done on the previous two hands. Careful analysis of table 96 discloses the nature of the relationship. The asterisks denote deviations of 3.5 standard errors or more. Without exception, the significant deviations are lower frequencies for the dealer getting a natural after having gotten a natural on one or both of the previous two hands. Thus this test is sensitive to the effects of removal of an ace and a 10 out of 416 cards! The other deviations from average follow no particular pattern, and are typical in size of the random fluctuations you expect to see in a sample of this size. Though the huge sample size makes this test extremely sensitive, there is no support for the notion that good hands beget good hands and bad hands beget

Table 95 Average Probabilities for Dealer's Hand, 817 natural 21 20 19 18

.048

17

.145

bust

.282

.073 .180 .139 .134

243

SHUFFLES

Table 96 Dealer's Hand Versus Two Preceding Dealer Hands previous hands

natural,natural natural,21 natural,20 natural,19 natural,18 natural,17 natural,bust 21 ,natural 21,21 21,20 21,19 21,18 21,17 21 ,bust 20,natural 20,21 20,20 20,19 20,18 20,17 20,bust 19,natural 19,21 19,20 19,19 19,18 19,17 19,bust 18,natural 18,21 18,20 18,19 18,18 18,17 18,bust 17,natural 17,21

17,20 17,19 17,18 17,17 17,bust bustnatural bust21 bu5t20 bust19 bust18 bust17 bustbust

.286 .283 .283 .283 .281 .282 .284 .283 .283 .282 .283 .282 .281 .280 .282 .281 .281 .282 .282 .282 .282 .283 .282 .281 .281 .282 .282 .281 .283 .282 .282 .282 .283 .282 .281 .282

17 .146 .145 .147 .146 .146 .147 .145 .145 .144 .145 .145 .145 .144 .145 .145 .145 .146 .145 .146 .146 .146 .145 .145 .145 .145 .145 .145 .146 .146 .145 .145 .146 .145 .145 .145 .146

18 .136 .139 .139 .139 .140 .139 .139 .139 .138 .140 .139 .139 .140 .140 .139 .139 .140 .139 .139 .140 .140 .141 .140 .140 .140 .140 .139 .140 .139 .138 .140 .138 .139 .139 .139 .140

.283

.145

.138

.283 .283 .282 .282 .281 .281 .282 .282 .281 .282 .281 .281

.145 .144 .145 .145 .145 .146 .145 .146 .145 .145 .145 .145

.139 .139 .139 .139 .140 .140 .139 .139 .139 .139 .139 .140

bust

19 .135 .134 .134 .134 .134 .134 .133 .134 .134 .133 .134 .133 .134 .133 .135 .133 .134 .133 .133 .133 .134 .133 .133 .133 .133 .134 .134 .133 .134 .133 .134 .133 .133 .133 .133 .135

20 .180 .179 .178 .179 .180 .179 .178 .181 .180 .180 .179 .180 .181 .180 .179 .178 .178 .181 .180 .179 .179 .178 .180 .180 .181 .179 .180 .179 .179 .181 .179 .181 .180 .181 .181 .180

21 .073 .075 .073 .073 .074 .072 .074 .072 .072 .073 .073 .073 .073 .073 .073 .074 .073 .073 .073 .073 .073 .073 .073 .073 .072 .072 .073 .072 .073 .073 .073 .073 .072 .072 .073 .073

.133

.180

.073

.048

.134 .133 .133 .133 .134 .134 .134 .134 .134 .134 .133 .134

.180 .181 .180 .180 .180 .180 .180 .179 .180 .180 .181 .180

.073 .073 .072 .072 .072 .073 .073 .073 .073 .072 .073 .073

.047 .048 .048 .048 .048 .047 .047 .048 .048 .048 .047 .048

natural

.045* .045* .046* .047 .045* .046 .046* .046 .048 .047 .048 .048 .048 .048 .047 .048 .047 .047 .048 .047 .047 .046 .048 .047 .047 .048 .047 .048 .046 .048 .047 .047 .048 .048 .048 .046*

244


bad hands. Thus there is no such thing as an honest dealer who can be relied upon to stay hot - or to stay cold.

Blackjack Without Shuffling The above section' shows that there is no such thing as streakiness if the cards are shufiled thoroughly. This section examines the opposite extreme - what happens if the cards are not shuffled at all. For cards new from the factory, shuflling is necessary. If a customer at the table is sophisticated enough to keep track of the location of certain cards or groups of cards, shuffling of used cards is necessary. But aside from that, does shuffling accomplish anything, or is it merely superstition and show? Argument For Shuffling Thorough shuflling is necessary to get an arrangement of cards that behaves in a random fashion. Argument Against Shuffling The cards are placed in the discard pile in an order different from that in which they are dealt. This guarantees that even if the cards are not shuffled, there is no way that the same cards could be dealt to the same players shoe after shoe. It is true that this reordering of cards is not random, and that some card combinations will occur with frequencies different from those if the cards were randomly shuffled. This might result in players winning slightly faster or slower than if the cards were shuffled, but it should not cause winning rounds to be followed by winning rounds or losing rounds to be followed by losing rounds. Reordering By Dealing Different dealing procedures result in different reorderings of cards. The reordering procedure used in

SHUFFLES

245

this study is called "lay and pay." An alternative procedure is called "pick and pay." Naturals (unless the dealer has a chance of a tying natural) and busted players' hands go to the discard pile first. In lay and pay, the dealer's hand is next on the discard pile. Then come unbusted player's hands, with first base on the bottom. If pick and pay were used, the dealer's hand would go on top. In the discard pile, the order of the cards in each player's hand is first card received on top and last card received on the bottom. The order of discards of the dealer's hand is hole card on the bottom, upcard next, and hit cards on top in the order received, with the last card received on top. For example, if there is one player against the dealer and neither takes a hit, cards ordered 1-2-3-4 before the deal are 1-3-2-4 on the discard pile. For two such rounds in succession, cards ordered 1 through 8 before the deal are 5-7-6-8-1-3-2-4 on the discard pile. For one round on which the dealer and player each take two hits, cards ordered 1 through 8 before the deal are ordered 1-3-5-6-8-7-2-4 on the discard pile. Pair splits are similar to multiple players in that the hand closer to first base is finished first, and ends up on top of the other hand in the discards. If there is only one player, that player splits a pair, and each split hand requires a third card, cards ordered 1 through 8 before

the deal are 3-7-8-1-5-6-2-4 on the discard pile. Three players, none of whom take hits, would cause cards 1 through 8 to be 3-7-2-6-1-5-4-8 on the discard pile. Purpose The purpose of this study was to find out how a player would fare shoe after shoe if the cards were shuffled only when taken new from their boxes.

246


Methodology Six decks were shufiled, and one card was burned. There was one player betting $100 and playing one hand of basic strategy against Atlantic City rules (dealer stands on soft seventeen, double after splits, and no resplits). The player's cards and the dealer's cards were placed in the discard pile in the same order as lay and pay in a casino. This procedure continued round after round until four and a half decks had been used, at which time the round was finished, the unused cards were placed on top of the discards, and the pack was cut. The cut was random and no closer than one deck to either end of the pack. For each shoe the study kept track of the amount the player won or lost. Then it compared performances on consecutive shoes. Results The results for more than a million shoes (over 44 million rounds) are shown in table 97. The first column is the win in the previous shoe. The second column is the number of shoes with a given win in the previous shoe. The next column is the number of hands in those shoes. After that comes the most important column, average win per hand. The last column is the standard error applicable to each average win. Table 97 shows that shuffling is necessary. The rearrangement of cards for the situation tested, namely one player, lay and pay, and six decks, results in the player being 0.70% to 0.75% better off than if the cards were shuffled. The average payback for one player using basic strategy is 100.25%, and the standard error on this number is 0.02%. A player advantage due to not shuffling may sound farfetched, but John Gwynn and Arnold Snyder came up with the same result in a later study

247

SHUFFLES

Table 97 One Player, $100 Bets, Six Unshuffled Decks, Lay and Pay win, prev shoe -$1000 or less -$950 or -$900 -$850 or -$800 -$750 or -$700 -$650or-$600 -$550 or -$500 -$450 or -$400 -$350 or -$300 -$250 or -$200 -$150or-$100 -$50 or zero $50or$100 $150 or $200 $250 or $300 $350 or $400 $450 or $500 $550 or $600 $650 or $700 $750 or $800 $850 or $900 $950 or more total

KEY se

Standard error.

shoes 97,298 25,579 30,380 34,765 39,169 43,416 47,343 50,427 52,938 54,461 55,377 55,068 54,826 52,354 49,316 45,228 41,782 37,387 32,832 28,524 115,784 1,044,254

hands avg win se 4,147,719 $0.14 $0.06 1,090,278 0.60 0.11 1,294,846 0.32 0.10 1,482,673 0.35 0.09 1,669,770 0.46 0.09 1,850,967 0.43 0.08 2,018,584 0.26 0.08 2,150,228 0.28 0.08 2,257,191 0.24 0.08 2,321,709 0.25 0.07 2,361,017 0.18 0.07 2,347,264 0.16 0.07 2,337,371 0.37 0.07 2,231,678 0.21 0.08 2,102,559 0.29 0.08 1,927,914 0.18 0.08 1,781,407 0.29 0.09 1,593,599 0.18 0.09 1,399,614 0.28 0.10 1,216,007 0.30 0.10 4,936,597 0.14 0.05 44,518,992 $0.25 $0.02

248


("Ruffied by the Shufl1e," Blackjack Forum, March 1990, pages 5-12). Table 97 indicates that winning or losing in the previous shoe does not foretell winning or losing in this shoe, even if the cards are not shuffled and the number of players does not change. Differences between win rates of the various lines of table 97 are so small that they probably are random. There is no such thing as streakiness if the cards are never shuffled. Streakiness does not exist at the two extremes, when cards are well-shuffled and when they are not shuffled. Since streakiness does not exist at both extremes, it does not exist between the extremes. That is, streakiness does not exist, regardless of the amount of shuffling.

Card Combinations Without Shuffling This section explains how not shuffling except when the cards are new can give the player an edge. Methodology This study consists of 500,000 consecutive shoes. The details were the same as in the previous study. At the conclusion of each shoe, the order of the cards in the discard pile was examined. The frequencies with which each pair of consecutive cards occurred were tallied. Then the study was repeated, and the frequencies with which each pair of alternate cards occurred were tallied. Results Table 98 contains the expected frequencies for any two cards, including subsequent and alternate, for thorough shuftles. This table is based on mathematics, not simulation.

249

SHUFFLES

Table 98 Any Two Cards from WellShuffled Six-Deck Shoes first card

ace 2

3 4 5

6 7 8 9 10

ace

2

.074 .077 .077 .077 .077 .077 .077 .077 .077 .077

.077 .074 .077 .077 .077 .077 .077 .077 .077 .077

relative frequency of another card 7 4 5 6 3

.077 .077 .074 .077 .077 .077 .077 .077 .077 .077

.077 .077 .077 .074 .077 .077 .077 .077 .077 .077

.077 .077 .077 .077 .074 .077 .077 .077 .077 .077

.077 .077 .077 .077 .077 .074 .077 .077 .077 .077

.077 .077 .077 .077 .077 .077 .074 .077 .077 .077

8 .077 .077 .077 .077 .077 .077 .077 .074 .077 .077

9

10

.077 .077 .077 .077 .077 .077 .077 .077 .074 .077

.309 .309 .309 .309 .309 .309 .309 .309 .309 .305

Table 99 contains the actual relative frequencies for subsequent cards for the sample of 500,000 unshuftled shoes. The simulations were played out for one player head-on against the dealer. Table 100 contains the actual relative frequencies for alternate cards for the same sample of 500,000 unshuffled shoes. There are many differences between tables 98, 99, and 100. All of these differences are statistically significant due to the large sample sizes. The differences arise because the play of the hands is not random. For example, aces as the first two cards to the player are always split, which causes consecutive aces to be less common with unshuflled cards than with shuflled cards, .058 to .074. Since split aces receive one and only one more card, aces with a card between them occur more often with unshuftled than shutlled cards, .082 to .074.

250


Table 99 Consecutive Cards From Unshuffled Six-Deck Shoes first card

ace 2 3 4 5 6 7 8 9 10

relative frequency of subsequent card ace

.058 .082 .082 .081 .081 .081 .078 .079 .079 .075

2 .078 .073 .084 .084 .083 .081 .075 .075 .074 .073

3 .078 .084 .072 .083 .082 .081 .075 .075 .074 .074

4 .077 .082 .082 .076 .081 .080 .075 .076 .075 .074

5 .076 .081 .080 .080 .077 .080 .075 .076 .075 .075

6 .077 .080 .079 .079 .079 .071 .076 .077 .076 .076

7 .079 .076 .076 .077 .077 .078 .069 .078 .078 .078

8 .081 .075 .075 .076 .076 .077 .079 .062 .079 .080

9 .081 .073 .074 .074 .074 .076 .079 .080 .070 .080

10

.315 .294 .296 .290 .291 .296 .319 .323 .321 .314

In general, the difference between well-shuffled cards and unshuffled cards is that if the cards are not shuffled, high cards tend to follow high cards and low cards tend to follow low cards.

Shuffle Every Fifty Shoes Peter Griffin asked what would happen with a shuffle every 50 shoes. So I ran another million shoes with a shuffle after every 50, and the result is a payback of 100.23% (with a standard error of 0.02%). This 100.23%

is not significantly different from the 100.25% of the previous study.

Other Numbers of Players I also ran simulations of 300,000 shoes each with various numbers of players and a shuffle every 50 shoes. With two players, the result was a payback of 99.90% with a standard error of 0.03%. With three players the

251

SHUFFLES

Table 100 Alternate Cards From Unshuffled Six-Deck Shoes first card

ace

relative frequency of card after subsequent card 2 3 4 5 6 7 8

9

ace 2 3 4 5 6

.082 .073 .077 .075 .077 .081 .077 .081 .077 .080

7

.m.m.m.~.m.m

.m .m m ~ .m .m

8 9 10

.078 .078 .075

.077 .077 .078

.073 .074 .074 .074 .076 .077 .080 .081 .081 .079 .076 .075 .076 .080 .080 .078 .076 .075 .080 .077 .079 .078 .076 .076 .080 .079 .076 .078 .076 .076

.m.~.~.~

.076 .076 .075 .075 .076 .076

m

.~

.076 .076 .077 .076 .076 .076 .076 .076 .076

10

.077 .321 .074 .301 .074 .302 .075 .301 .076 .302

.076 .078 .077 .078 .079 .079

.~ .~

.311 .312 .310

result was a payback of 99.73% with a standard error of 0.05%. With four players, the result was a payback of 99.72% with a standard error of 0.07%. With five players, the payback was 99.67% with a standard error of 0.08%. Thus the payback for blackjack without shuftling varies with the number of players at the table; more players means a lower player edge.

252


APPENDIXES A & B

The appendixes are not recommended reading. You do not need them to play blackjack to win. They are for those readers who want a wider range of numbers than is presented in the earlier chapters. Each consists of four tables, identified by letter and number. The letter designates the counting system. Appendix A - high-low Appendix B - halves The number differentiates the number of decks and the dealer action on soft seventeen. 1 - one deck, dealer hits soft seventeen 2 - four decks, dealer hits soft seventeen 3 - one deck, dealer stands on soft seventeen 4 - four decks, dealer stands on soft seventeen Each table is spread over three pages. The first page is a table covering hitting or standing, doubling down, and pair splitting. The second page covers hitting or standing and doubling down by two-card combinations. The third page covers surrender by two-card combinations. Then comes a blank spacer page.

ApPENDIX A

253

To use the numbers in the table, first drop the fraction from your count per deck. Examples: a count per deck of 5.2 rounds to 5, 2.8 rounds to 2, -2.3 rounds to -2, and -1.9 rounds to -1. A number in a table means different things, depending on where you are in the table. It means stand, double down, split, or surrender if your count per deck is the number or more. If your count per deck is lower than a number in the table, then hit, do not double down, do not split, or do not surrender. If a number is followed by an asterisk, the usual meaning is reversed. The upper limit on early surrender is only if you cannot both surrender and buy insurance. If you can do both on the same hand, then ignore the numbers that are followed by an asterisk in the surrender tables. For example, look at table A4 for early surrender of 8-8 against ace. The table says -14/12*. If the count per deck is less than -14, do not surrender, and play your hand according to the advice elsewhere in the table. If the count is between -14 and +12, surrender. If the count per deck is above +12: Surrender if you can both surrender and buy insurance, but if doing one means you cannot do the other then buy insurance and do not surrender. These tables are my own independent work. Any resemblance between them and the work of Julian Braun or anyone else is fortunate. My method of calculation is to start with one or four decks, and remove the known cards (the dealer's upcard and your two cards, if known). I then add cards in proportion to the values assigned to them by the counting system (for negative values that means subtracting cards) to achieve a desired count per deck. Then I assume that I have an infinite number of those resulting packs. I find the expected value of each possible play for each hand.

254


When the best play changes as I change the count per deck, I have found a strategy number for that hand. These tables differ from earlier editions of this book. The change was increasing all the negative numbers by 1 because in the 1981 and earlier editions of Professional Blackjack, strategy numbers were presented using a different assumption as to how to round count per deck.

Appendix A The set of tables in Appendix A contains the decision rules for the high-low count (2,3, 4, 5, 6 = +1; 7, 8, 9 = 0; 10, ace = -1). The four tables in this part of the appendix cover both dealer actions on soft seventeen for one and four decks. (text continues on page 270)

KEY TO APPENDIXES A AND B Stand. Double down. db Hit. h hI7 Dealer hits soft seventeen. Do not double. See the hI- row for the number saying h/whether to hit or to stand. hlsp Either hit or split, but never stand. spl Split. Sp/Either split or stand but never hit. number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. Reverse the meaning. Split only if the count per deck * is less than the number in the table. With 4-4, splitting is better than doubling down only ** if doubling is allowed after splitting.

ApPENDIX

255

A

Table Al High-Low, One Deck, HI7 Player's Hand

Deae ~ r'.s Upeard 7 6

2

3

4

5

soft 18 hard 17 "0 hard 16 jen hard 15 ~ hard 14 :E hard 13 hard 12 11 10 9 eu 8 :c:s 7 0 6 "0 5

-16

-18

-15

-16

-14

-9 -5 -3 0 3 -11 -8 1 14

-10

-12 -8 -3 0 -13 -10 -2 5 11 15 15

-13 -9 -7 -4 -1 -14 -11 -4 2 9 11 12

-13 -11 -9 -7 -2 -15 -13 -5 2 8 12 13

11 12 16 h h -8

ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2 ace-ace 10-10 9-9 8-8 7-7 6-6 5-5 4-4** 3-3 2-2 ace-ace 10-10 9-9 8-8 7-7 6-6 5-5 4-4** 3-3 2-2

10 9 0 0 15 18 12 11 -13 11 0 spl -14 0

9 4 -1 -4

5 1 -10 -14 -7 -6 -4 -1 -15 5 -4 spl spl -6

4 0 -15 -15 -15 -10 -6 -3 -16 5 -5 spl spl -7

15 16 18

4 5 5 -14 9 -1 spl -16 -1

7 2 -10 -8 -3 -3 -1 1 -15 7 -2 spl spl -4

-12 13 6 spl spl 0*

18 4 5 -14 9 -3 spl -16 -5

11 -1 0 -15 7

7

8 9 -13 11 -2 spl -14 -3

spl spl -7

-15 5 -5 spl spl -9

8 -5 -7 -16 5 -6 spl spl -10

13* spl -12 13 3 spl spl 11*

17 -2 -1

4 -6 -4

0 -9 -6

-2 -11 -9

-6 -11 -10

spl spl

0

:c:s 0 "0

Q)

:0

= 0 "0 0

~ .t:

0.. en Q)

:0

=

0 "0

.d ~ .~

.-:::

~ en

-8 -4 -1 2 -12 -9 0 9 17

4

-8

-4

-4 -4

-8 3 14

8

9

10

ace

-16 -19 10 12 h h h -5 -3 8

h -19 5 8 h h h -3 0

h 1 5 h h h -3 4

17 -5 3

-10 20 -14 spl

-10

-10

-8

-12 11*

6*

0 -2

-10

-10

-8

-12 18*

8*

0 -2

-10 20 -14 spl -1

0 11

5 9 15 h -1 3

256


Table At (Continued) Player's Hand

"tj

s:: ~

~

Uj

-.... ~

:.a "'0 ... ~

CJ I

0

~ ~

(l)

:0 ::s 0 "'C

...

"'0 Cd CJ I

0

~ ~

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6

2

3

4

Dealer's Upcard 5 6 7

-7

-8

-8

-9 spl -7 -6 -5 -4 -3 -4 h/sp 0 0 -3 -3 4 2 1 1 h/sp

-7 -6 -5 -6 spl -3 -2 -5 -5 1 0 0 0 h/sp

-7 -7 -7 spl -4 -3 -6 -6 0 -1 -1 -1 h/sp

-1 -3 -3 -3 h/sp

-11 -12 -13 -12 -9 -9 -9 -9 -1 0 0

-12 -13 -14 -14 -10 -11 -11 -11 -3 -2 -2

-13 -14 -15 -15 -11 -11 -12 -12 -4 -4 -4

-14 -15 -16 -15 -12 -13 -12 -14 -5 -4 -6

9 9 10 17 17

6

3 2 2 9 8 11 12

3 1 1

spl -6 -5 -4 -2 -3 -3 h/sp 1 1 -2 -2 5 3 3 2 h/sp

9-2 8-3 7-4 6-5 8-2 7-3 6-4 5-5 7-2 6-3 5-4

-10 -11 -11 -12

6-2 5-3 4-4 5-2 4-3 4-2 3-2

14 14 16

-8 -8 -8 -8 0 1 1

-10 -11 spl -9

-8

5 6 12 11 15 15

8

9

-17 -12 12 13 spl 11 11 18 h h h 20 h h h h h h h h h

-18 -17 8 9 sp/7 8 11 h h h 13 h h h h h h h h h

-8

-5 -5

-3 -3

-9

-5

-3

-8

-5 -2 -4

-3 -1 0 0 -1

-12 -11 spl

-11 -11 spl

13 19 spl

-10 -9

-12 -11 -10 -9 -9 -9 spl -6 -5

11 14 17 15 19 19 spl h h h h h h h h h

-8

-8 -8

8 7 12 13

-7

-5 -6 -6 -6 2 3 2 14 14 14

-3 -3

7 8

8

10

ace -5

-4 4 0 sp/4 4 4 h h h 0 h h h h h h h h h

4 3 h/sp 4 4 4 10 10 8 2 18 17 12 10 h h h 20 h

-2 -4 -5 -1 3 3 5 4

0 -1 -2 -1 3 2 3

3

257

ApPENDIX A

Table At (Continued) Player's Hand

~

(1)

--0

= (1) ~ ~ ~

::s ~

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 6-2 5-3 4-4 6-ace 5-2 4-3 5-ace 4-2 3-3 4-ace 3-2 3-ace 2-2

early 10 ace 4 8

16* 13*

-4 -4 -1 -3 -3 -1 0 0 0 -5 5 5 1 1 10 7 7 7 8

-16 -18/17* -18/10* -15 -15/19* -14/17*

18 14 13 15 11 11 16 12 16

-13 -13/20* -14/19* -19/17* -11 -10/20* -13/20* -14/19* -7/20* -10/19* -10/19* -9/19* -8/19* -214* -2/4* 0/3* -7/12* -9/13* -6/15* -6/15* -3/13* -1/10*

Dealer's Upcard early & late 7 8 9

10

18 16

12 15

11 14

12 13

6 6

2 2 5

11 12 13 14 14 14

6

2 3 4 8 8 7 2 16 17 11 10

-1 -1 1 0 0 1 4 4 3 -1 10 11

19 19

7 9 13 1313 8 19 19 13 12

15 15 16 19

6 6 17 13 13 14 15

19 18 18 15 16 19 17

late ace

-3/17* -3/10* -3 -1/13* 1/9* 4 5/15* 4/11* -3/8* 16

258

PROfESSIONAL BLACKJACK

259

ApPENDIX A

Table A2 High-Low, Four Decks, HI7 Player's Hand


2

3

4

soft 18 hard 17 "'t;j hard 16 hard 15 :1= hard 14 ...t:l hard 13 hard 12 11 10 9 0 :c 87 0 6 ~ 5

-14

-15

-14

ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2 ace-ace 10-10 9-9 8-8 7-7 6-6 5-5 4-4** 3-3 2-2 ace-ace 10-10 9-9 8-8 7-7 6-6 5-5 4-4** 3-3 2-2

10 8 0 1 15 18 14 12 -11 11 -1 spl -8 1

7 -12 8 -2 spl -10 -1

8 7 -11 11

18 3 3 -12 8

12 0 0 -13

-13

6

5

-2

-4

spl -10 -2

spl -11

-5 spl -13 -6

-6 spl -14

-8

spl -16 -10

1 -7 -7

-1 -9 -9

-11 -11

~

:c

=

0

"0

0

=§ 0 "0 0

~ .~

~ til

0

:c::s 0

~

':5 .~

.~

~ (/.)

-15

9

10

-15

h

h

ace h

-4 -9 -5 -3 0 3 -11 -8 1 13

-10 -7 -5

-11

-12

-8 -6

-9

-14 -11

-7

-9

-2

-3 0 -13 -10 -2 5 12 15 15

-4

-7

-1 -14 -11

-3 -16 -13

-4

-6

3 9 11 12

1 8 12 13

3 -6 -7 -3

5 1 -8 -10 -6

0

-4

1 3 -13

6

-1 0 -13 5

4 0 -13 -14 -13 -10 -5 -2 -14 4

-3 spl -12 -3

-4

-6

spl -14 -5

spl -16 -7

1 -12 -9 0

9 16 20 20

=

0

-15

8

14 0 -2

8 5

-2 -3 3 6

6

-4 6 -3 -5

6

8 -2

-4

9 10 17 h h -9

-6 3 14

7 10 h h h

5 8 h h h

0 4 h h h

-6 -4

-4

-4

-1

4

3 5 9 15 h 0 3

-8

-4

7

14 17 20

-9 13 6 spl spl

-8

-7

20 -8 spl

17*

6*

-1

13* spl -9 13 3 spl spl

-8 20

-7

-8

-4

-8 spl 5/19*

-9 18*

8*

1 -1

spl spl

4 5

-9

2

7 -5 -7 -14 4

-8

-6

260


Table A2 (Continued)

~

d

~ ...., tt.:I ........... ....,

:2 ~

...... ~

(.) I

0

~ ....,

(l.)

:c 0='

"'C

] (.) I

0

~ ....,

Player's Hand

2

3

4

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6

-9 -9 spl -6 -6 -5 -3 -3 -3 h/sp 0 0 -1 -1 3 3 3 2 h/sp

-10 -10 spl -7 -6 -6 -5 -4 -5 h/sp -1 -1 -2 -2 2 1 1 1 h/sp

-11 -11 spl -8 -8 -7 -6 -6 -6 h/sp -3 -3

-12 -12 spl -9 -9 -8 -7 -7 -7 h/sp

-4 -4

-11 -11 -11 -11 -8 -8 -8 -8 0 1 1 13 13 13

-12 -12 -12 -12 -9 -9 -9 -9 -1 0 0 9 9 9 16 16 20 20

9-2 8-3 7-4 6-5 8-2 7-3 6-4 5-5 7-2 6-3 5-4 6-2 5-3 4-4 5-2

4-3 4-2 3-2


8

9

10

ace -4 -4 4 4 h/sp 5 5 5 10 10

0 0 0 0 h/sp

9 10 spl 10 11 11 17 18 18 spl h h h h h h h h h

8 8 spl 10 10 11 h h h 20 h h h h h h h h h

5 5 sp/8 8 9 h h h h h h h h h h h h h

1 0 sp/4 4

-5 -5 -1 -1 -1 -1 h/sp

-14 -14 spl -11 -11 -11 -9 -9 -9 h/sp -6 -6 -7 -7 -2 -3 -3 -3 h/sp

-13 -13 -13 -13 -10 -10

-14 -14 -14 -14 -11 -11

-15 -16 -16 -16 -13 -13

-8 -9 -9 -9 -5 -6

-6 -6 -6 -6 -3 -4

-4 -4 -4 -4

-4 -4 -4 -4

-1 -1

4 4

0 0 0 0 3 3

-10

-11

-13

-6

-3

-1

4

3

-10 -2 -2 -2 5 5 5 12 11 15 15

-11

-13 -6 -6 -6 1 1 1 8 8 12 13

-6 3 3 3 14 14 14

-4

-1

4

3

-4 -4

-4 -4 -4 3 3 2 9 8 11 12

7 7 7

4 h h h 11 h h h h h h h h h

9 8 16 16 15

14 h h h h h

261

ApPENDIX A

Table A2 (Continued) Player's Hand

.....

cu

"Q

= cu ..... .....

..... ::s CIJ

early 10 ace

Dealer's Upcard early &late 7 8 9

10

10-7 9-8 10-6 9-7 8-8

5 6 -5 -5 -2

18* 17* -17 -18 -18/14*

10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 6-2 5-3 4-4

-2 -2 -2

-14 -14 -13 -13 -13 -13 -14 -11 -10 -11 -11 -7

6-ace 5-2 4-3

18 14 14 15 11 11 16 12

-2114*

19 16 16 20 16

15

0/11*

20

5-ace 4-2 3-3 4-ace 3-2 3-ace 2-2

0 0 0 -1 3 4 3 2 8 7 7 7 8

-8 -8

13 13 13 10 10 11 11 16 16 16

4 4 6 7 7 12 12 12 11 20 20 19 18

0 0

11 12 -2 -2

7

0

2 2

0 0 0

3 7 7 6 5 14 14 12 12

20

-8 -8 1/4* 1/4* 2/4* -6/12* -7/12* -4/15* -4/15*

3 3 3 2 8 9 7 7 14 13 13 13 14

19 19 20 20

late ace

-4/16* -4/14* -1/16* 0/15* 0/14* 4/16* 4/15* 4/14* 2/13*

262


263

ApPENDIX A

Table A3 High-Low, One Deck, S17 Player's Hand soft 18 hard 17 ~ hard 16 jrJ'J hard 15 ~ hard 14 .d hard 13 hard 12 11 10 9 Q.) 8 :0 7 0 6 ~ 5

2

3

4

-17

-18

-15

-8 -4 -2 0 4 -11 -8 1 14

ace-8 ace-7 Q.) :0 ace-6 ace-5 0 ace-4 ~ ace-3 ace-2 ace-ace 10-10 Q.) ,..... 9-9 .0 8-8 ~ 7-7 6-6 5-5 4-4** .~ ~ 3-3 2-2 ace-ace Q.) 10-10 ::s 9-9 0 8-8 ~ 7-7 .s 6-6 .~ 5-5 .~ 4-4** ~ rJ'J 3-3 2-2

=

=

~

t')

:c

9 17

=

ace-9

-10 -6 -4 -1 2 -12 -9 0

10

9

9

4 -1 -4 4 5 5 6 -14 9 0 spl -17 -1

1 0 15 19 13 11 -13 11 0 spl -16 1

18

-12 -8 -6 -3 0 -13 -10 -2 5 11 15 16 7 2 -10 -8 -3 -3 0 1 -15 7 -2 spl spl -4

Dealer's Upcard 5 6 7 -16 -13

-9 -7 -4 0 -14 -11 -4 3 9 11 12 5 1 -10 -14 -7 -6 -3 -1 -15 5 -4 spl spl -6

8 -4 -4 -15 5 -5 spl spl

-5

12 -1 0 -15 7 -4 spl spl -7

5 -7 -4

0 -9 -6

4

8 9 -13 11 -1 spl -16 -2

5 -14 9 -2 spl -17

18 -4 -2

-17 -11 -10 -8 -4

0 -15 -12 -5 2 9 15 16 5

0 -13 -15 -15 -10 -6

11 12 16 h h -8 -6 3 14

8

9

10

-16 -19 10 12 h h h -5 -3 8

h -19 5 8 h h h -3 0

h 1 5 h h h -3 4

13 20 h 0 4

-10 20 -14 spl

-10

-10

-7

-12 11*

6*

1 -16

-10

-10

-7

-12 18*

8*

1 -16

ace -1 -6 8

9

15 16 18

-3 -16 5

-3 spl spl -7 18 -10

-12 13 6 spl spl 0*

-9

-8 -16 5 -5 spl spl -11

13* spl -12 13 3 spl spl 11*

-2 -11 -9

-4 -13 -11

spl spl

-10 20 -14 spl -1

0 11

264


Table A3 (Continued) Players Hand

~

= ~

~

fn

.......... ~

:.E ~

r-c

~ (,.) I

0 ~ ~

Q.)

:c:s 0 "'0

"CS

r-c

~ (,.) I

0

~

~

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 9-2 8-3 7-4 6-5 8-2 7-3

2

-7 -7 spl -5 -5 -4 -1 -2 -2 h/sp 2 2 -1 -2

6 3 3 2 h/sp -10 -11 -11 -11 -8 -8

6-4

-8

5-5 7-2 6-3 5-4 6-2 5-3 4-4 5-2 4-3 4-2 3-2

-8 0 1 1 14 14 16

3

-8 -8 spl -7

-6 -5 -4 -3 -4 h/sp 0 0 -3 -3 4 2 1 1 h/sp -10 -12 -13 -12 -9 -9 -9 -9 -1

0 0

9 9 10 17 17

4

Dealers Upcard 7 5 6

8

9 -18 -17 8 9 sp/7 8 11 h h h 13 h h h h h h h h h

-10 -10 spl -8 -7 -7

-11 -11 spl -10 -9 -7

-9 -9 spl -10 -9 -8

13 19 spl 11 14 17

-17 -12 12 13 spl 11 11 18

-6

-7 -6 -7 spl -4 -3 -6

-7 -7 -7 spl -3 -3

-6

-6 0 -1 0 0 h/sp

15 19 19 spl h h h h h h h h h

h h h 20 h h h h h h h h h -5 -5 -5 -5 -2 -4 -3 -3 7 8 8

-5 -5 spl -2 -1 -4 -5 2 0 0 0 h/sp -12 -13 -14 -14 -9 -11 -11 -11 -3 -2 -2 6 5 6 12 11 15 16

0 -1 -1 -1 h/sp

-6

-13 -14 -15 -15 -11 -11 -12 -12 -4 -4 -4 3 2 2

-14 -15 -15 -15 -12 -12 -12 -13

-7 -8 -9 -8 -5 -6 -6

-6

2 3 2 14 14 14

9

9 9

8 11 12

-4

-6 3 1 2

15 16

-6

-3 -3 -3 -3 -1 0 0 -1

10

ace

4 0 sp/4 4 4

-7 -6 8 8 h/sp 9 8 9

h h h 0 h h h h h h h h h

15 14 12 7 h h 17 14 h h h h h

-2

0

-4

0

-5

0

-1 3 3 5 4

4 3 4 4

0

265

ApPENDIX A

Table A3 (Continued)

.... Q) "t:S

= ..... Q)

.....

.....

::s rn

Player's Hand

early 10 ace

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 6-2 5-3

4 15* 8 12* -4 -13 -4 -14/20* -1 -12/6* -3 -12 -3 -12 -1 -10/19* 0 -10 0 -9 0 -10 -5 -14/19* 5 -7 5 -6 1 -9 1 -10 10 -4/20* 7 -fJ/20* 7 -fJ/20* 7 -fJ/19* 8 -fJ/20*

Dealers Upcard early & late 7 8 9

12 13 11 12 13 14 14 14

19 19

18 16 6 6

6 7 9 13 13 13 8 19 19 13 12

12 15 2 2 5 2 3 4 8 8 7 2 16 17 11 10 15 15 16 19

10 11 14 -1 -1 1 0 0 1 4 4 3 -1 10 11 6 6 17 13 1.3 14 15

4-4 6-ace 5-2 4-3 5-ace 4-2 3-3 4-aoo 3-2 3-ace 2-2

18 14 13 15 11 11 16 12 16

-4/12* -5/13* -2118* -2118*

0/16* 1fT*

19 18 18 15 16 19 17

late ace

0 0/18* 0 1 3/17* 5 7 6/20* 0/16* 15 18 10 10

266


267

ApPENDIX A

Table A4 High-Low, Four Decks, 817 Player's Hand soft 18 hard 17 hard 16 hard 15 hard 14 :i= ,d hard 13 hard 12 11 10 9 Q) 8 ..D 7 0 6 ~ 5 ace-9 ace-8 ace-7 Q) ace-6 :c ace-5 =' 0 ace-4 ~ ace-3 ace-2 ace-ace 10-10

1 ....-4

2

4

-15

-15

-15

-15

-17

-8 -5 -3 0 3 -11 -8 1 13

-11 -7 -6 -3 0 -13 -10 -2 5 12 15 15 6 3 -6 -7 -2 0 1 3 -12 6 -3 spl -13 -3

-12 -9 -7 -4 -1 -14 -11 -4 3 9 12 12 5 1 -8 -10 -6 -4 -1 0 -13

-13 -9 -7

2

-10 -6 -4 -1 2 -12 -9 0 9 16 20 20 8 5 -2 -3 4 7 7 7 -12 8 -2 spl -11 0 19 3 3 -12 8 -3 spl -11 -4

13

8 7 -11 11 -2 spl -9 -1

15 0 -3

6 -4 -5

=

Q)

=§ 0

9-9 8-8

7-7 6-6 ~ 5-5 4-4** .~ ~ 3-3 en 2-2 ace-ace Q) 10-10 9-9 =' 0 8-8 ~ 7-7 ,.c:: 6-6 .~ 5-5 .-= ,..... 4-4** Q.. en 3-3 2-2 '1j

:c ~


3

10 8 0 1 15 18 14 13 -11 11 0 spl

-9

5 -4

-4 0 -15 -13

-6 1 9 15 16 4 1 -10 -13 -12 -9

9

10

ace

-15

h

h

7 10 h h h -6 -4 7

5 8 h h h -4 -1

0 4 h h h -4 4

1 -6 8 10 14 20 h 1 4

-8 20 -8 spl

-7

-8

-3

17*

6*

3 -11

-7

-8

-3

8*

3 -11

14 17 20

-4 -1 -14 4

-4

spl -14 -5

spl spl -7

8

18

-2

-9

0 -12 6 -5 spl -13 -6

-4 -13 5 -6 spl -14 -8

-9 -14 4 -6 spl spl -10

1 -7 -7

-1 -9 -9

-4 -14 -12

a

9 10 17 h h -9 -6 3 14

8

-9 13 6 spl spl

13* spl -9 13 3 spl spl

-8 20 -8 spl 5/19*

spl spl

4 5

-9

-9 18*

268



~

= ~

... ~

til '-...

:E ~

.....

~ C,,) I

0

~

~

Q,) .....c

J:j

::s

0 "0 ~

a C,,) I

0 ~ ~

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5

8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 9-2 8-3 7-4 6-5 8-2 7-3 6-4 5-5 7-2 6-3 5-4 6-2 5-3 4-4 5-2 4-3 4-2 3-2

2

3

4

-8 -8 spl -5 -5

-9 -9 spl -6 -6 -6 -4 -4 -4 h/sp -1 -1 -2 -2 2 1 1 1 h/sp

-11 -11 spl -8 -7 -7 -6 -5 -5 h/sp

-12 -12 -12 -12 -9

-13 -13 -13 -13 -10 -10 -10 -10 -2 -2 -2 5 5 5 12 12 15 15

-5 -3

-3 -3 h/sp

0 0 0 0 4 3 3 3 h/sp -11 -11 -11 -11

-8 -8 -8

-9

-8

-9

0 1 1 13 13 13

-1 0 0 9 9 9 16 16 20 20

-9

-3 -2 -3 -3 0 0 0 0 h/sp


-12 -12 spl -9 -9 -8 -7 -7 -7 h/sp -4 -4 -5

-12 -12 spl -9 -9 -9 -7 -7 -7 spl -4 -4 -5

-5

-5

0 -1 -1 -1 h/sp

0 0 0 0 h/sp

-14 -14 -14 -14 -11 -11 -11 -11 -4 -4 -4 3 3 3

-15 -15 -15 -15 -12 -13 -12 -13 -6 -5 -6 1 1 1 9

9 9 11 12

9 15 16

8

9

10

ace -6 -5


8 8 spl 10 10 11 h h h 20 h h h h h h h h h

5 5 sp/8 8 9 h h h h h h h h h h h h h

1 0 sp/4 4 4 h h h 11 h h h h h h h h h

9 9 h/sp 10 9 10 14 14 14 12 'h h 19 19 h h h h h

-8

-6

-9

-6 -6

-4 -4 -4 -4 -1 -1 -1 -1

-4 -4

1 1 1 1 4 4 4 4

-9 -9

-5 -6 -6

-6 3 3 3 14 14 14

-6 -3 -4 -3 -4 7 7 7

-4 -4 4

4 4

4

269

ApPENDIX A


.... (I)

"'d

= .... (I)

.... .... ::s tI)

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 6-2 5-3 4-4 6-ace 5-2 4-3 5-ace 4-2 3-3 4-ace 3-2 3-ace 2-2

early 10 ace 5 6 -5 -5 -2 -2 -2 -2 0 0 0 -1 3 4 3 2 8 7 7

7 8

17* 17* -14 -14 -14/12* -11 -11 -10 -9 -9 -9 -10 -7 -7 -8 -8 -4 -5 -5 -5 -5

18 14 14 15 11

-1/17*

11

-1/17*

Dealer's Upcard early &late 7 8 9

13 10

4 4

10 11 11 16 16 16

6 7 7 12 12 12 11 20 20 19 18

13 13 0 0

7 2 2 3 7 7 6 5 14 14 12 12

20

-2112* -3/12*

10 11 12 -2 -2 0

0 0

0 3 3 3 2 8 9 7 7 14 13 13 13 14

19 19

19 20 20

16 16

16 12

0/16*

20 16

15

2111*

20

late ace

-1 0 1 2 2 6 7 6 5 15 16 14 14

270


Appendix B This set of tables contains the decision rules for the halves count (2, 7 = +0.5; 3, 4, 6 = +1; 5 = +1.5; 8 = 0; 9 = -0.5; 10, ace = -1). The four tables in this part of the appendix cover both dealer actions on soft seventeen for one and four decks. (text continues on page 286)

KEY TO APPENDIXES A AND B Stand. db

Double down.

h hI7 h/-

Hit. Dealer hits soft seventeen. Do not double. See the h/- row for the number saying whether to hit or to stand.

h/sp

Either hit or split, but never stand.

spl

Split.

Sp/Either split or stand but never hit. number Stand (or double or split) at a count per deck equal to or greater than the number; hit (or do not split) at a count per deck less than the number. Reverse the meaning. Split only if the count per deck * is less than the number in the table.

**

With 4-4, splitting is better than doubling down only if doubling is allowed after splitting. If doubling is not allowed after splitting, then doubling on 8 is more profitable than splitting 4-4 no matter how high the count. The numbers for splitting 4-4 with no double after split are for use if you are not allowed to double on a total of eight.

ApPENDIX

B

271

Table Bl Halves, One Deck, HI7 Player's Hand

"0

~

+::s

:E

Cl)

:0 ::s 0 "0

Cl)

:0 ::s 0 "0

Cl)

:0 ::s 0 "0 0

~ .~

0.. en Cl)

:D ::s 0 "0

..d ~ .~

.t::

Q.. en

soft 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12 11 10 9 8 7 6 5 ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2 ace-ace 10-10

9-9 8-8 7-7 6-6 5-5 4-4**

3

4

-18

-18

-15

-16

-15

-9 -6 -3 0 3 -12

-10 -6 -3 -1

-12 -8 -5 -3 0 -13 -10 -2 5 11 14 15 6

-13 -9 -6

-14 -12 -9

-4

-6

0 -14 -11

-3 -16 -13

-4

-6

3 9 11 12 5 1 -8 -12

2 9 12 13 4 0 -13 -14 -15 -11 -8 -5 -17 5 -5 spl -18 -8

-9 0 14

9 8 0 0 15 17 14 11 -15 11

-2 spl -12 0

-12 -9 0 9 16 18 20 8 4 0 -3 4 4 5 5 -14 9 -1 spl -11 -1

2 -8 -6 -2 -3 -1 0 -15 6 -3 spl -14 -4

7 -15 11 -3 spl -12 -4

11 -1 0 -15 6 -5 spl -14 -7

19 -7 -3

4 -8 -3

0 -9

9

2-2 ace-ace 10-10 8-8 7-7 6-6 5-5 4-4** 3-3 2-2

2

17 4 4 -14 9 -3 spl -11 -5

3-3

9-9


2

-6

-6 -6

-4 -2 -15 5

-4 spl -15 -7 9

12 12 14 h h -8 -5 4 17

-12 14 5 spl spl 0*

spl -15 -10

10* spl -12 14 3 spl spl 12*

-3 -12 -8

-7 -13 -11

spl spl

-4 -15 5

-6

9

10

ace

h

h

12 14 h h h -5 -3 9

5 9 h h -4 -1

0 4 11 h h -3 4

h -5 3 4 7 12 h -1 3

-11

-11

-10

-8

-15 spl

-14 14*

5*

-1 -2

-11

-11

-10

-8

-15 spl 0

-14 18*

8*

-1 -2

20

16 14 16 10

10 -6 -6 -17 5 -7 spl -18 -12

-9

8 -15

-3/12* 8

272


Table HI (Continued) Players Hand

"'C

= td

~

tI)

.......... ~

:c "'C td """"

(,) I

0 ~ .....

Q,)

:c:s 0

""0

"0 td """" 0

I

0

.....~

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 9-2 8-3 7-4 6-5 8-2 7-3 6-4 5-5

7-2 6-3 5-4 6-2 5-3 4-4 5-2 4-3 4-2 3-2

2

3

-8 -8

-8 -8

spl

spl

-6 -6

-6

-4 -2 -3 -3 h/sp

0 0

-7 -5 -3 -3 -3 h/sp 0

0

4

-10 -11 . spl -8 -8 -6 -5 -5 -5 spl -2 -2 -3 -4 1 -1 0 0 h/sp

2 h/sp

-2 -2 3 1 0 2 h/sp

-12 -12 -12 -12 -9 -9

-12 -12 -12 -12 -9 -9

-13 -13 -13 -13

-9

-9

-10

-8 0

-8

-10 -3 -2 -2 5 6 5 12 10 14 15

-2 -2

5 2

2

1 1 13 15 16

-1 0 0

8

10 9 16 16 18

20

-10 -10

Dealer's Upcard 7 6

5

-12 -11 spl -9 -10 -7

-11

14

-12

20

spl

spl

-12 -12 -10 -9 -8 -8 spl -6 -5 -7 -7 -2 -4 -4 -2 h/sp

12 15 18 12 17 16 spl h h h h h h h h h

-14 -14 -14 -14 -11 -11 -12 -11

-16 -16 -16 -15 -13 -13

-8

-4

-6

-4

-5 -6 2 1 1

-6 -6 -6 spl -3 -3 -4 -5 0 -2 -1 0 h/sp

-4 3 3 3 9 9

11 12

-8

-8 -8 -5 -5

8 -18 15 15 spl 13 12 h h h h 18 h h h h h h h h h

9

9

4

-5 -5 4 3 h/sp

0

sp/-

sp/4 3 4 13 12 10 0 h h h

8 7 12 h 20 17 12 h h h h h h h h h

h h h h h h

-6 -5 -5

-4

-3

-4 -3

-4

-5

-4

-3 -3

-3 0

-13

-6

-3

-1

-5 3 4 3 16 17 16

-3 7 8 10

-1

8 12 13

ace

8

-13

9

10

-4 -3 3 3 5 5

5 4 5 8 8 7 3 14 14 10 9 h 20 h h h -1 -1 -1

0 2 3 3 4

ApPENDIX

B

273

Table Bl (Continued) Player's Hand

;... Q)

"tj

= Q)

;... ;... ;...

::s

til

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 6-2 5-3 4-4 6-ace 5-2 4-3 5-ace 4-2 3-3 4-ace 3-2 3-ace 2-2

early 10 ace 4 7 -4 -4 -1 -3 -3 -1 0 0 0 -3 4 4 1 1 11 7 8 10 10

17* 13* -16 -17/18* -14/11* -15 -16/19* -13/19* -12 -12/20* -12/20* -17/19* -10 -10/20* -12 -13/20* -9 -11/20* -11/20* -10 -9/20* -2/4* -1/5* 0/3*

19 15 -8/12* 14 -10/13* 16 11 -6/16* 11 -6/16* 17 12 -3/14* 20 14 -2/11*

Dealer's Upcard early & late

7

13 13 12 11 14 14 13 13

17 17

8

9

10

18 14 7

11 12 1 1 5 2 2 4 6 6 5 2 13 13 9 9

11 14 -1 -1 2 0 0 1 3 4 3 0 9 9 6 6 20 13 15 18 18

6 7 7 9 11 11 11 8 15 15 12 11

13 14 17 18

19 18 20

20 19 19 15 16 19 17 19

late ace

-3/15* -2/9* -2/17* -1/11* 1/9* 3/18* 4/13* 3/11* -2/9* 12/18*

274


ApPENDIX

B

275

Table B2 Halves, Four Decks, HI7 Players Hand


2

3

4

soft 18 hard 17 ~ hard 16 hard 15 CIl ~ hard 14 :E hard 13 hard 12 11 10 9 Q) 8 :0 7 0 6 ~ 5

-15

-15

-15

-15

-16

-9 -6 -3 0 3 -12 -9 1 13

-11 -8 -5 -3 0 -13 -10 -2 5 11 14 15

ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2 ace-ace 10-10 9-9 8-8 7-7 6-6 5-5 4-4** 3-3 2-2 ace-ace 10-10 9-9 8-8 7-7 6-6 5-5 4-4**

10 7 0 1 14 17 15 13 -12 11 -1 spl -7 1

-10 -7 -4 -1 1 -12 -9 0 8 15 18 18 8 4 -1 -2 3 5 6 7 -12 8 -2 spl -8 0

6 3 -5 -5 -2 0 1 3 -13 6 -3 spl -10 -3

-13 -9 -7 -4 -1 -14 -11 -4 3 9 12 13 5 1 -7 -9 -6 -5 -2 0 -14 5 -4 spl -11 -5

-15 -12 -9 -6 -3 -16 -13 -6 1 9 13 14 4 0 -11 -13 -13 -11 -6 -3 -15 4 -6 spl -14 -8

8 6 -12 11 -3 spl -12 -2

17 3 2 -12 8 -4 spl -12 -4

12 0 0 -13 6 -5 spl -14 -6

9 -2 -3 -14 5 -6 spl -14 -8

-6 -15 4 -8 spl -17 -11

3-3

0

2-2

-2

6 -4 -4

1 -7 -6

-2 -10 -8

-7 -13 -10

!

=

Q)

:0

= 0

~

Q)

:0 ::s 0

~

0

.........

~

.~

"a CIl Q)

:0 ::s 0

~

.: ~

.~

.~

'a CIl

17

8

9

10

ace

-15

h

h

8 11 18 h h -7 -4 8

5 9 19 h h

0

-1

4 10 h h -4 4

h -4 3 5 8 13 h 0 3

-10 14

-9

-8

-9

-5

5

-9 spl

-10 spl

5*

2 -1

17 -9

-8

-9

-5

-10 spl

8*

1 -1

9 10 14 h h -9 -6 3 16

-4

15 15 17 14

spl spl

9 -6

8* spl -10 14 3 spl spl

-9 spl 2

spl spl

4

276


Table B2 (Continued) Player's Hand 10-7 9-8 10-6 9-7

8-8

"t:S

= ~

~

til

.......... ~

:a "t:S

.....

~ (,,) I

0 ~

~

Q)

:c::s

0 ""0

"t:S

..... ~

0

I

0 ~ ~

10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 9-2 8-3 7-4 6-5 8-2 7-3 64 5-5 7-2 6-3

5-4 6-2 5-3

4-4 5-2 4-3 4-2 3-2

2

3

3 3 h/sp

h/sp -1 -1 -2 -2 1 1 1 1 h/sp

-11 -11 spl -8 -8 -7 -5 -5 -5 h/sp -3 -3 -3 -3 0 0 0 0 h/sp

-12 -12 -12 -12 -9 -9

-12 -12 -12 -12 -9 -9

-9

-8 0 1 1 13 14 14

-9 -9 spl

-6 -6 -5 -3 -3 -3 h/sp 0 0 -1 -1 3

2

-10 -10 spl -7 -7 -6 -4 -4

4

-4


-12 -12 spl -9 -9 -9 -7 -6 -6 h/sp

-4 -4 -4 -4

-14 -14 spl -12 -12 -11 -9 -9 -9 h/sp

-6 -6 -7 -7 -3

-1 -1 -1 -1 h/sp

-3 -3 h/sp

-13 -13 -13 -13 -10 -10

-14 -14 -14 -14 -11 -11

-16 -16 -16 -16 -13 -13

-9

-10

-11

-13

-9 -1 0 0 8 8 8 15 15 18 18

-10 -2 -2 -2 5 5 5 11 11 14 15

-11 -4 -4

-13 -6 -6 -6 1 1 1 9 8 13 14

-4 3 3 3 9 9 12 13

-4

8

9

10 ace -4 -4 4 3 h/sp 5 5 5 8 8 8 7 13 13 12 12 h h h h h


9 9 spl 11 11 13 18 18 19 17 h h h h h h h h h

6 6 spl 9 8 10 20 20 19 17 h h h h h h h h h

1 0 sp/4 4 4 11 11 10 7 h h h h h h h h h

-9 -9 -9 -9

-7 -7 -7 -7 -4 -4 4 -4 8 8 8

-4 -4 -4 -4 -1 -1

-4 -4

-6 -6 -6 -5 3 3 3 16 16 16

-1

-1

-4 -4 4 4 4 4

0 0 0 0 3 3 3

3

ApPENDIX

B

277

Table B2 (Continued) Player's Hand 10-7

9-8

s-. Cl)

~

s::

Cl)

s-. s-. s-. ::s rn

10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5

early 10 ace 5 6

-5 -5

6-6

-2 -2 -2 -2 0 0 0 0 3 3 2 2 9 8 8 9 9

6-2 5-3 4-4 6-ace 5-2 4-3 5-ace 4-2 3-3 4-ace 3-2 3-ace 2-2

19 15 15 16 11 11 17 12 20 14

19* 18* -17 -17 -18/15* -14 -14 -13 -11 -11 -12 -13 -10 -10 -10 -11 -9 -9 -9 -9 -9 1/4* 1/4* 2/4*

Dealer's Upcard early &late 7 8 9 20 19 13 11

5 5

13 11 12 14 14 14

7 6 7 10 10 10 9 16 16 15 15

11 11 0 0 7 2 2 3 5 6 5 5 11 11 10 10

-7/12* -8/13* -4/16* -4/16* -2/15* -1/13*

19

10 12 12 -2 -2 0 0 0 0 3 3 3 2 7 8 7 7 17 15 16 16 16

20 20 19 16

18

16

19

20 17 19

late ace

-4/14* -4/13* 0/15* 0/13* 0/13* 3/15* 3/14* 3/13* 2/13* 12/15* 13/14* 11/14* 11/13*

278


ApPENDIX

B

279

Table B3 Halves, One Deck, S17 Players Hand

"'0

B (/)

:t= ..c::

Q)

.,D

::s

0 "'0

Q)

:0

='

0 "'0

Q)

:c::s 0 "'0 0

~

.-=:

2

3

4

soft 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12 11 10 9 8 7 6 5 ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2 ace-ace 10-10 9-9 8-8 7-7 6-6 5-5 4-4**

-18

-18

-15

3-3

9

-9 -5 -2 0 4 -12 -9 0 13

9 8 0 0 15 17 15 12 -14 11 -1 spl -12 0

-10 ~

-3 -1 2 -12 -9 0 9 16 19 20 8 4 0 -3 4 4 5 5 -14 9 -1 spl -12 -1

2-2 ace-ace Q) 10-10 ::s 9-9 0 8-8 "'0 ..c: ..... 7-7 6~

7 -14 11 -3 spl -12 -3

5-5 4-4** 3-3 2-2

20 -8 -3

5 -8 -4

(/)

:c .~

.-=:

~ (/)

-16

-17

8

9

10

-15

h

h

ace 0 ~

17 4 4 -14 9 -3 spl -12 -5

'a


-12 -8 -5 -2 0 -13 -10 -2 5 11 14 15 6 2 -7

-6 -2 -2 -1 0 -15 6 -3 spl -14 -4

-13 -9 ~

-4 0 -14 -11 -4 3 10 11 13 5 1 -8 -12

-6 -6 -4 -2 -15 5

-12 -10 -7

-4 0 -15 -13 -5 2 10 15 17 4 0 -10 -14 -14 -10 -7

12 12 14 h h -8 -5 4 17

~

-16 -8 -17 5

12 -1 0 -15 6 -4 spl -14 -7

10 -11 -4 -15 5 ~

~

spl -15 -10

spl spl -13

10* spl -12 14 3 spl spl 12*

0 -10

-2 -12 -8

-5 -16 -11

spl spl

~

-11

-11

-10

-7

-15 spl

-14 14*

5*

0 -18

-11

-11

-10

-7

-15 spl 0

-14 18*

8*

0 -18

-4

-4 -12 14 5 spl spl 0*

-4

8 9 12 17 h 0 4

5 9 20 h h

16 14 16 10

-17 5 -3 spl sol -8

spl -15

-1

0 4 11 h h -3 4

12 14 h h h -5 -3 9

-3/12* 8

280


Table B3 (Continued) Player's Hand

"0

r::

tU ..... ""'....."' CI)

:.a

"0 ~

tU

(,J I

0

~ .....

cu

::c = 0

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 9-2 8-3 7-4 6-5 8-2 7-3

2

-7

-8 spl -5 -6

-4 -2 -2 -3 h/sp 1 1 -1 -1 5 2 3 3 h/sp -12 -12 -12 -12 -9

-8


8

-10 -10 spl -10 -10 -8

14 20 spl 12 15 18

-18 15 15 spl 13 12 h

-4 -4

-7 -6 -6 spl -3 -3 -5 -5

h h h 18 h h h h

1 0 0 0 h/sp

0 -2 -1 0 h/sp

0 -2 -1 0 h/sp

12 17 16 spl h h h h h h h h h

-12 -12 -12 -12 -9 -9

-13 -13 -13 -13 -10 -10

-14 -14 -14 -14 -11 -11

-15 -15 -15 -15 -13 -12

-8

-6 -5 -5 -5 -3 -3 -3 -3 7 8 10

3

-8 -8 spl -6 -6 -4 -3 -2 -3 h/sp 0 0 -2 -2 4 1 1 2 h/sp

4

-10 -10 spl

-8 -8 -6 -5 -4 -5 h/sp -2 -1 -3 -3

-12 -11 spl -9 -10 -7 -6 -6 -6 h/sp -3 -3

"'0

6-4

-9

-9

-10

-12

-12

...

5-5 7-2 6-3

-8 0

-8

-10 -3 -2 -2 5

-11

-13 -6 -5 -6 2 1 1 10 9 15 17

"0 tU

(,J I

0

~ .....

5-4 6-2 5-3

4-4 5-2 4-3

4-2 3-2

1 1 13 15 16

-1 0 0 8 10

9 16 16 19 20

6 5 12 11 14 15

-4 -4 -4 3 3 3 10 9 11 13

-8 -8

-8 -5 -5 -6 -5 3

4 3 16 17 16

h h h h h

9

9 8 sp/8 7 12 h 20 17 12 h h h h h h h h h -4 -4 -3 -4 -3 0 -1

-1

10

4 0 sp/4 3 4 13 12 10 0 h h h h h h h h h -3 -4 -4 -3 3 3 5 5

ace -7 -7 8 8 h/sp 9 7 9 12 12 10 8 19 18 15 13 h h h h h 0 0 0 0 3 4 4 5

ApPENDIX

B

281


early 10 ace

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 6-2 5-3 4-4 6-ace 5-2 4-3 5-ace 4-2 3-3 4-ace 3-2 3-ace 2-2

4 17* 7 12* -4 -13 -4 -14/20* -1 -1217* -3 -12 -3 -13 -1 -10 0 -9 0 -9 0 -10 -3 -12 4 -7 4 -6 1 -8 1 -10 11 -5 7 -7 8 -7 10 -6 10 -7

$-c Q)

"'0

=

Q)

$-c $-c $-c

::1 en

19 15 14 16 11 11 17 12 20 14

Dealers Upcard early &late 7 8 9

13 13 12 11 14 14 13 13

17 17

18 14 7 6 7 7 9 11 11 11 8 15 15 12 11

11 12 1 1 5 2 2 4 6 6 5 2 13 13 9 9

10 11 14 -1 -1 2 0 0 1

13 14 17 18

3 4 3 0 9 9 6 6 20 13 15 18 18

-4/13* -5/14* -3/19* -3/19*

19

20 19 19 15

18

16

0/18*

20

19 17

0/14*

19

late ace

0 0/17* 0 0/17* 4/17* 4 6 5/17* 1/17* 12 14 9 8/18*

282

PROFES SIONALB LACKJACK

ApPENDIX

B

283

Table B4 Halves, Four Decks, 817 Player's Hand soft 18 hard 17 "0 hard 16 hard 15 ~ hard 14 :E hard 13 hard 12 11 10 9 (1,) 8 :0 7 :s 0 6 '"tj 5 ace-9 ace-8 ace-7 Q) ace-6 ace-5 =' 0 ace-4 "0 ace-3 ace-2 ace-ace 10-10 (1,) 9-9 8-8 :s 0 7-7 "0 6-6 ~ 5-5 ~ 4-4** .~ 0.. 3-3 en 2-2 ace-ace (1,) 10-10 :0 9-9 =' 0 8-8 '"tj 7-7 ..c ....- 6-6 .~ 5-5 .~ 4-4** 0.. 3-3 2-2

B C/.)

:c

:c

C/.)


2

3

4

-16

-16

-15

-16

-19

-9 -5 -3 0 3 -12 -8 0 13

-10 -6 -4 -1 2 -12 -9 0 8 15 18 19 8 5 -1 -2 3 6 7 7 -12 8 -2 spl -8 0

-11 -8 -5 -3 0 -13 -10 -2 5 11 14 15 6 3 -4 -5 -2 0 1 3 -13 6 -3 spl -10 -3

-13 -9 -6 -4 -1 -14 -11

-13 -10 -7

spl -11 -5

0 -16 -13 -6 1 10 16 18 4 1 -8 -12 -12 -10 -5 -2 -15 5 -4 spl -16 -7

-12 11 -2 spl -9 -2

18 3 3 -12 8 -3 spl -9 -4

13 0 0 -13 6 -4 spl -11 -6

10 -2 -3 -14 5 -6 spl -12 -8

-18 -8 -15 5 -6 spl -16 -11

18 0 -2

7 -5 -4

1 -8 -6

-2 -10 -8

-5 -18 -10

10 7 0 1 15 17 16 14 -12 11 -1 spl -8 1 9 6

-4 3 9 12 13 5 1 -7 -9 -6 -5 -1 0 -14 5

-4

-4

9 10 14 h h -9 -6 3 16

8

9

10

ace

-15

h

h

8 11 18 h h -7 -4 8

5 9 19 h h -1

0 4 10 h h -4 4

1 -6 8 9 12 17 h 1 4

-9

-8

-9

-4

-9 spl

-10 spl

5*

3 -16

17 -9

-8

-9

-4

-10 spl

8*

3 -16

-4

15 15 17 14

-10 14 5 spl spl

8* spl -10 14 3 spl spl

-9 spl 2

spl spl

4

284



"t:S

= cd

~

VJ ........... ~

:.a "t:S

..... cd

(,) I

0

~

~

Q)

:0 ::s

0 "'t:S "'0 ..... cd

(,) I

0

~

~

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 9-2 8-3 7-4 6-5 8-2 7-3 64 5-5 7-2 6-3 5-4 6-2 5-3 4-4 5-2 4-3 4-2 3-2

2

3

4


8

9

10

ace

-5 -5 -5 h/sp -2 -2 -3 -3 0 0 0 0 h/sp

-12 -12 spl -9 -9 -8 -6 -6 -6 h/sp -4 -4 -4 -4 -1 -1 -1 -1 h/sp

-13 -13 spl -10 -10 -9 -7 -7 -7 h/sp -4 -4 -4 -4 0 -1 -1 0 h/sp


9 9 spl 11 11 13 18 18 19 17 h h h h h h h h h

6 6 spl 9 8 10 20 20 19 17 h h h h h h h h h

1 0 sp/4 4 4 11 11 10 7

h h h h h h h h h

-6 -6 8 8 h/sp 10 9 9 12 12 12 11 17 17 17 16 h h h h h

..a

-12 -12 -12 -12 -9 -9

-13 -13 -13 -13 -10 -10

-14 -14 -14 -14 -11 -11

-16 -16 -16 -16 -13 -13

-9 -9 -9 -9 -6

-4 -4 -4 -4 -1 -1

-4 -4 -4 -4 4 4

1 1 1 1 4 4

-8

-9

-10

-11

-13

-6

-1

4

4

-8 0 1 1 13 14 14

-9 -1 0 0 8 8 8 15 15 18 19

-10 -2 -2 -2 5 5 5 11 11 14 15

-11 -4 -4 -4 3 3 3 9 9 12 13

-13

-5 3 3 3 16 16 16

-7 -7 -7 -7 -4 -4 4 -4 8 8 8

-1

4

4

..a ..a

-9 -9 spl -6 -6 -6 -4 -4 -4 h/sp -1 -1 -1 -1 2 1 1 2 h/sp

-12 -11 -11 -12 -9

spl -5 -5 -5 -3 -3 -3 h/sp 0 0 0 0 4 3

3 2 h/sp

-11 -11 spl -8 -8 -7

-6 -6 -6 1 1 1 10 10 16 18

-6

ApPENDIX

B

285


10-4 Q.)

"'0

c:

Q.) 10-4 10-4 10-4

=' en

10-7 9-8 10-6 9-7 8-8 10-5 9-6 8-7 10-4 9-5 8-6 7-7 10-3 9-4 8-5 7-6 10-2 9-3 8-4 7-5 6-6 6-2 5-3 4-4 6-ace 5-2 4-3 5-ace 4-2 3-3

4-ace 3-2 3-ace 2-2

early 10 ace 5 6 -5 -5 -2 -2 -2 -2 0 0 0 0 3 3 2 2 9 8 8 9 9

19 15 15 16 11 11 17 12 20 14

18* 17* -14 -14 -13/12* -11 -11 -10 -9 -8 -9 -9 -7 -7 -7 -7 -5 -5 -5 -5 -5

Dealer's Upcard early &late

7

8

9

10

13 11

20 19 5 5

11 11 0 0 7 2 2 3 5 6 5 5 11 11 10 10

12 12 -2 -2 0 0 0 0 3 3 3 2 7 8 7 7 17 15 16 16 16

13 11 12 14 14 14

7 6 7 10 10 10 9 16 16 15 15

-3/13* -3/13* -1/18* -1/18*

19 18

0/18*

19

1/15*

20 20 19 16 16 20 17

19

late ace

-1 0 1 1 2 5 6 5 4 12 12 11 11

286


APPENDIXC

This appendix lists some simulation results generated by Blackjack Count Analyzer. The tables in this appendix show the frequencies with which certain counts per deck appear, and the win rates at those counts. Each count per deck is truncated before being inserted into a table. For example, a hand with a count per deck of +4.8 appears as part of the +4 group, and a hand with a count per deck of -4.8 appears as part of the -4 group. There are three zero groups. The columns labeled 0 mean exactly zero. The columns labeled +0 mean greater than zero but less than +1. The columns labeled -0 mean negative but greater than -1. For single deck there is no -0 or +0 because it is impossible to get such a count per deck with the high-low system. Each table in this appendix contains three rows. The fIrst row is the frequency with which each count per deck occurred in the simulation. The second row is the win rate for each count per deck as determined by the simulation. The third row is the standard error appli-

ApPENDIX

C

287

cable to the win rate. The win rates are estimates, and the standard errors define the precision of those estimates. Smaller standard errors are due to larger sample sizes. The win rates for the one, two, and six-deck benchmarks are based on sample sizes exceeding 600 million, so those standard errors are small. The other win rates are based on smaller samples - 100 million or more - and thus have larger standard errors. For a given count per deck, the win rate is fairly constant from table to table - if the same rules and counting system are used. For the benchmark rules, each count per deck is worth about 0.5% up to the threshold for buying insurance, above which it is worth more than 0.5%. For double exposure, table C7 shows each count per deck to be worth about 0.7%. Penetration is 52 cards for tables C 1, C2, C6, C7, and C8. It is 26 cards for C3, 62 cards for C4, and 42 cards for C5. For a given set of rules, the different win rates associated with different numbers of decks and different levels of penetration are due to the frequencies with which high counts occur. For example, for the six-deck benchmark, counts per deck of +6 and over occur 2.49% of the time. With 62-card penetration, counts per deck of +6 and above occur only 2.04% of the time. To generate more such tables as are shown in this appendix, get Blackjack Count Analyzer.

288


Table Cl Frequencies and Win Rates 6-Deck Benchmark Count per deck

-6& under -5 f% ev

se

-4

-3

-2

-1

-0

0

+0 +1

+6& +2 +3 +4 +5 over

2.43 1.51 2.38 4.04 6.69 11.56 17.18 9.24 16.77 11.32 6.48 4.03 2.37 1.51 2.49 95.77 96.68 97.08 97.61 98.21 98.67 99.09 99.38 99.70 100.22100.76101.35101.91102.50104.21 .03.04 .03 .03 .02 .02 .01 .02 .01 .02 .02 .03 .04 .05 .04

Table C2 Frequencies and Win Rates 2-Deck Benchmark Count per deck ~&

under -5 f%

ev

se

~&

-4

-3

-2

-1

-0

0

+0 +1

+2 +3 +4 +5 over

2.20 1.75 2.78 4.44 7.42 11.64 9.17 23.82 9.06 10.56 6.41 4.42 2.51 1.65 2.18 95.91 96.78 97.24 97.75 98.24 98.76 99.22 99.53 99.95 100.38101.02101.61102.18102.67104.10 .03 .04 .03 .02 .02 .02 .02 .01 .02 .02 .02 .03 .03 .04 .04

Table C3 Frequencies and Win Rates I-Deck Benchmark Count per deck

-6 & under -5 f%

ev se

-4

-3

-2

-1

5.52 3.67 3.37 5.12 5.69 10.12 95.94 97.11 97.50 98.02 98.48 99.18 .02 .03 .03 .02 .02 .02

-0

0

0

36.63 99.81 .01

+0 +1 0

+6& +2 +3 +4 +5 over

9.80 4.73 4.82 2.87 2.03 5.64 100.84101.43102.18102.68103.49105.09 .02 .02 .02 .03 .04 .02

ApPENDIX

C

289

Table C4 Frequencies and Win Rates 6-Deck, Shuffle at 62 Count per deck ~&

under -5 f%

ev

sa

~&

-4

-3

-2

-1

-0

0 +0 +1 +2 +3 +4 +5 over

2.00 1.40 2.30 3.99 6.72 11.76 17.60 9.35 17.15 11.50 6.53 3.96 2.30 1.39 2.03 95.73 96.68 97.15 97.68 98.18 98.66 99.09 99.41 99.69 100.19100.64101.37101.93102.49104.05 .07 .08 .07 .05 .04 .03 .02 .03 .03 .03 .04 .05 .07 .09 .08

Table C5 Frequencies and Win Rates 6-Deck, Shuffle at 42 Count per deck

-6& under -5 f%

ev sa

-4

-3

-2

-1

-0

+6& 0 +0 +1 +2 +3 +4 +5 over

2.94 1.59 2.49 4.13 6.70 11.40 16.56 9.16 16.15 11.18 6.49 4.11 2.50 1.60 3.02 95.80 96.83 97.19 97.75 98.10 98.64 99.14 99.23 99.73 100.21100.93101.33101.97102.59104.41 .10 .14 .11 .09 .07 .05 .04 .05 .05 .06 .07 .09 .12 .15 .11

Table C6 Frequencies and Win Rates Halves, 6-Deck Count per deck ~&

under -5 f% ev

se

~&

-4

-3

-2

-1

-0

0 +0 +1 +2 +3 +4 +5 over

2.78 1.62 2.54 4.14 6.76 11.48 17.96 6.05 17.60 11.24 6.63 4.16 2.56 1.63 2.85 95.81 96.78 97.20 97.75 98.14 98.65 99.09 99.40 99.68 100.15100.72101.29101.83102.35104.19 .05 .06 .05 .04 .03 .02 .02 .03 .02 .02 .03 .04 .05 .07 .05

290


Table C7 Frequencies and Win Rates Double Exposure Benchmark Count per deck

-6& under -5 f% ev

se

-4

-3

-2

-1

-0

0

+6& +0 +1 +2 +3 +4 +5 over

2.46 1.52 2.40 4.02 6.70 11.64 16.98 9.41 16.72 11.40 6.48 3.98 2.31 1.51 2.47 94.43 96.20 96.93 97.55 98.24 98.94 99.45 99.79 100.09100.72101.38102.05102.58103.24104.16 .06 .08 .06 .05 .04 .03 .02 .03 .02 .03 .04 .05 .06 .08 .06

Table C8 Frequencies and Win Rates Double Exposure Halves Count per deck

-6& under -5 f% ev

se

-4

-3

-2

-1

-0

0

+6& +0 +1 +2 +3 +4 +5 over

2.83 1.64 2.59 4.22 6.76 11.46 17.95 6.23 17.41 11.25 6.59 4.13 2.51 1.62 2.81 94.58 96.35 96.98 97.76 98.18 98.82 99.47 99.78 100.11100.72101.34102.08102.54 103.29104.90 .09 .12 .10 .08 .06 .05 .04 .07 .04 .05 .06 .08 .10 .13 .10

291

APPENDIX D

This appendix contains simulation data showing how often you are likely to face each situation. There are two ways to do it. One way is to look at only the initial hands. Another way is to look at how often you encounter each situation in a casino; that is, recognize that you might have to make more than one decision on one hand. For example, if you are dealt 4-3 and hit with 7, you have to make a decision with a total of seven and also with fourteen, and if you hit your fourteen you may have to make yet another decision before you are done with the hand. I am presenting the data both ways. Tables D1 and D3 recognize only the decision you had to make with your initial total of seven, and tables D2 and D4 recognize all the decisions you had to make on the hand. Tables D1 and D2 use one deck, and tables D3 and D4 use six decks. All four tables use benchmark rules and basic strategy. There is no double counting; for example, 4-4 shows up as a pair of 4s only, and is not included in the number for hand total of eight.

292


The dealer is assumed to check the hole card, so all decisions against a 10 and ace are based on the dealer not having a natural. If the dealer has a natural, the player hand totals are not added in to any of the tables in this appendix. After you double down and receive another card you do not have a decision to make, so your total after doubling down is not included in tables D2 and D4. Likewise with your final hands after splitting a pair of aces. The data in the tables in this appendix are expressed as frequencies per 100,000 hands. They are simulation results from a modified version of Blackjack Count Analyzer, for samples of more than 100 million hands each. The simulations played two hands at a time of basic strategy with benchmark rules, including shuffling if 52 or fewer cards remain for the six-deck numbers, and if 26 or fewer cards remain for the single deck. Use of different rules, different numbers of simultaneous hands, or different shuflle points could get slightly different numbers for tables D2 and D4. Individual frequencies might be slightly different (higher or lower by 1 per 100,000) for other 100-millionhand samples. The amount of sampling error can be seen by comparing the numbers in tables D 1 and D2 that ought to be the same. For example, player 10-10 against 2: table Dl says 723 and table 02 says 724. That difference is due solely to the two tables being generated from different 100-million-hand samples. Most of the deviation of those numbers from 36 is due to having a fixed shuffle point rather than dealing a constant number of rounds between shuffles. Dealing another round is more likely from a pack poor in face cards. This effect is much more pronounced in single

ApPENDIX

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293

deck than in multiple-deck games, and is easy to see from the frequencies of 10-10 in table Dl. You might expect each frequency against small upcards to be (16xI5x4)/(52x51x50) or 724 per 100,000 hands, but the actual frequencies range downward from 724. The only hands missing from table D1: Dealer naturals were 4808 per 100,000, of which 176 tied player naturals. Dealer naturals were equally split between 10 up and ace up. The only hands missing from table D3: Dealer naturals were 4744 per 100,000, of which 225 tied player naturals. Dealer naturals were equally split between 10 up and ace up. Lawrence Revere presents first-two-card data for one deck on page 7] of Playing Blackjack as a Business. Julian Braun presents first-twa-card data for four decks on page 82 of How to Play Winning Blac/fjack. Chambliss and Roginski present first-twa-card data for six decks on page 42 of Playing Blackjack in Atlantic City. All three of these data sets apply only if a fixed number of rounds are dealt between shuftles. For example if you face a shuffle after every round, the data of Revere, Braun, and Chambliss/Roginski describe the frequency with which each situation arises. (text continues on page 298)

294


Table Dl Frequencies (per 100,000) of Initial Hands, 1 Deck, S17 Player's Hand A-A

10-10 9-9 8-8 7-7 6-6

5-5 4-4 3-3 2-2 aoo-10 ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14 hard 13 hard 12 11 10 9 8 7 6 5

2

3

4

36 723 36 36 36 36 37 37 37 19 388 97 97 98 98 98 98 98 74 386 386 485 485 583 585 683 586 368 270 271 172 173 74 74

36 722 36 36 36 37 37 37 19 37 386 97 97 97 97 97 98 73 98 385 387 482 484 582 583 585 658 367 269 270 171 173 99 74

36 720 36 36 36 36 37 18 37 37 387 97 97 97 97 97 73 98 98 385 386 482 482 581 486 656 656 365 269 269 197 172 74 99


36 720 36 36 36 36 18 37 37 37 386 97 97 96 97 73 97 97 98 384 384 482 482 483 557 655 656 366 293 269 171 172 98 99

36 717 36 36 36 18 36 37 37 37 384 96 97 97 73 97 97 97 98 383 384 481 385 555 556 655 679 365 268 268 171 196 98 98

36 716 36 36 18 36 36 36 37 37 384 96 97 72 97 97 97 97 98 383 383 384 456 554 578 651 654 364 267 268 195 196 98 98

8

9

10

A

36 716 36 18 36 36 36 37 37 37 384 96 72 97 97 97 97 97 98 382 286 455 479 553 553 651 652 364 266 292 195 195 98 98

36 714 18 36 36 36 36 37 36 37 384 72 96 96 97 96 97 97 98 286 382 455 455 551 553 651 651 362 291 291 194 195 98 97

139 2291 131 131 132 133 133 134 134 135 1347 360 360 361 362 362 362 363 364 1313 1312 1666 1667 2024 2025 2387 2387 1416 1064 1067 711 715 358 358

12 514 24 24 25 25 25 25 25 25 200 49 49 49 49 49 49 49 50 266 267 332 332 398 400 465 465 262 197 198 132 132 66 66

ApPENDIX

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295

Table D2 Frequencies (per 100,000) of Decisions, 1 Deck, 817 Player's Hand A-A

2

3

4

37 10-10 724 9-9 39 8-8 39 7-7 39 6-6 37 5-5 37 4-4 37 3-3 37 2-2 19 BJ 388 soft 21 not BJ 46 soft 20 150 soft 19 161 soft 18 157 soft 17 133 soft 16 122 soft 15 109 soft 14 103 soft 13 74 hard 21 79 hard 20 80 hard 19 488 hard 18 554 hard 17 712 hard 16 687 hard 15 793 hard 14 764 hard 13 879 hard 12 706 11 432 10 321 9 300 8 185 7 178 6 75 5 75

36 722 40 39 39 40 37 37 19 37 387 40 143 155 152 142 123 112 79 98 85 86 495 562 688 713 784

37 721 39 39 39 40 37 18 40 40 387 29 130 146 139 126 125 86 111 105 28 31 440 508 628 611 708 610 818 789 452 343 322 228


36 719 39 39 39 40 18 37 40 40 386 9 111 125 122 114 83 98 105 105 29 32 442 506 624 610 607 776 645 795 772 774 771 439 454 330 370 310 327 192 203 180 1~O 190 100 79 105 74 112 112

36 717 39 39 39 19 37 37 40 40 385 9 110 125 123 86 107 98 105 105 29 33 439 503 628 502 675 640

769 799 451 345 322 198 216 105 112

36 715 36 39 19 36 36 37 40 40 383 50 168 169 138 137 131 110 110 104 371 360 835 876 927 894 942 868 912 827 462 337 316 221 216 105 112

8

9

10

A

36 715 39 19 36 36 36 36 37 37 383 47 176 140 158 139 128 112 103 97 365 355 877 763 976 924 934 853

36 714 19 39 36 36 36 37 37 37 383 62 160 177 157 139 128 112 103 97 369 374 772 919 993 910 950 867 899 810 451 350 326 213 205 99 98

139 2292 131 143 132 133 133 133 134 135 1347 334 681 663 587 520 479 420 386 364 1445 1777 3158 3401 3773 3464 3599 3291 3390 3059 1731 1282 1196 780 747 364 359

12 513 24 26 24 24 25 25 25 25 201

884

797 453 323 327 213 204 100 98

44

91 87 78 68 64

55 52 49 362 363 640 678 729 666 688 628 645 591 319 237 220 144 138 67 67

296


Table D3 Frequencies (per 100,000) of Initial Hands, 6 Decks, S17 Player's Hand A-A

10-10 9-9 8-8 7-7 6-6 5-5

4-4 3-3

2-2 ace-10 ace-9 ace-8 ace-7 ace-6 ace-5 ace-4 ace-3 ace-2 hard 19 hard 18 hard 17 hard 16 hard 15 hard 14

hard 13 hard 12 11 10 9 8 7 6 5

Dealer's Upcard

2

44

3 44

4 44

5 44

6 44

7 44

8 44

9 44

727 728 727 726 725 726 726 726 44 44 44 44 44 44 44 40 44 44 44 44 44 44 41 44 44 44 44 44 44 40 44 44 44 44 44 44 40 44 44 44 44 44 44 41 44 44 44 44 44 44 41 44 44 44 44 44 44 41 44 44 44 44 45 44 41 44 44 44 44 44 44 44 368 367 367 368 368 368 367 366 92 92 92 92 93 92 92 88 92 92 92 92 92 92 88 92 92 92 92 93 92 88 92 92 92 92 92 92 89 92 92 92 92 92 92 88 92 92 92 93 92 92 88 92 92 92 92 92 92 89 92 92 92 92 92 92 88 93 92 92 92 92 92 91 368 367 369 368 366 367 366 351 367 368 366 368 368 366 351 367 460 461 459 460 459 444 455 455 460 460 461 459 444 454 459 455 552 553 552 536 547 548 547 548 551 644

629 365 273 272 181 181 89 89

10

A

164 2619 162 162 162 162 162 162 163 163 1345 340 339 342 341 341 341 341 341 1338 1340 1679 1677 2016

28 506 30 31 30 31 30 31 31 31 244 61 60 61 61 61 61 61 61 254 254 319 318 382

552

537

548

547

551

630 640 364 273 273 181 180 92 89

547

641 641 365 273 272 185 181 89 92

639 641 365 277 272 181 181 92 92

548 2018

383

640 643 364 272 272 180 184 92 92

639 640 365 273 272 184 184 92 93

640 638 2356 639 640 2360 364 363 1358 272 277 1016 277 276 1017 185 184 680 185 185 680 92 92 340 93 92 340

446 446 254 190 190 127 127 64 63

ApPENDIX

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297

Table D4 Frequencies (per 100,000) of Decisions, 6 Decks, S17 Player's Hand A-A

10-10 9-9 8-8 7-7

2 44

3 44

4 44

Dealer's Upcard 5 6 7 44 44 44

727 727 728 726 51 51 51 51 51 51 52 51 52 51 51 51 6-6 44 52 51 52 5-5 44 44 44 40 4-4 44 44 40 44 3-3 44 41 52 51 2-2 41 44 51 52 BJ 368 367 368 368 soft 21 not BJ 48 42 29 20 soft 20 168 141 126 118 soft 19 161 155 142 133 soft 18 156 150 137 128 soft 17 134 142 126 119 soft 16 121 122 118 106 soft 15 109 110 103 100 soft 14 99 95 108 108 soft 13 88 92 100 100 hard 21 95 87 32 32 hard 20 93 91 36 36 hard 19 575 487 435 434 hard 18 577 555 501 501 hard 17 728 678 622 620 hard 16 700 702 604 601 hard 15 802 779 690 672 hard 14 789 769 672 652 hard 13 887 851 820 810 hard 12 787 772 797 788 11 464 453 469 468 10 341 351 363 367 9 315 323 335 336 8 201 209 224 220 7 190 190 205 205 6 91 96 96 100 5 89 89 109 108

8 44

9

10

A

44

164 2620 162 188 162 162 162 163 163 163 1345 312 633 640 554 499 451 406 367 340 1408 1762 3130 3431 3726 3446 3551 3269 3337 3018 1677 1250 1150 755 717 352

28 505 30 35 31 30 31 30 30 31 244 55 113 114 99 89 80 72 65 61 351 358 620 674 727 671

726 725 725 727 51 44 51 47 51 51 47 51 51 47 44 44 47 44 44 44 44

44 44

44

51 52 368 10 108 124 119 105 102 93 101 100 32 37 434 501 616 581 678 647 780 783 468 364 335 219 209 100 108

51 52 368 50 161 164 155 133 125 108 107 100 377 376 826 863 991 894 937 850 914 831 473 355 326 216 208 100 108

44 44 44 44

367 48 168 157 151 135 122 110 98 92 369 365 850 835 969 900 927 853 873 789 459 334 313 205 195 95 92

44 44 44 44

366 59 174 173 150 135 122 110 99 92 373 373 836 902 985 911 941 867 886 801 457 338 311 205 195 95 92 340

688

634 646 585 314 233 216 141 134

66 64

298


APPENDIX E

Over the years I have made considerable use of the appendix tables in Edward O. Thorp's Beat the Dealer. I remember the first time I used them: I had just learned about a casino offering surrender, and I wanted to know which player totals have an expectation of worse than -50%. I am reminded of this every time I use those tables because I underlined all player totals of fifteen and sixteen against the dealer's 10 in my copy of Beat the Dealer. The purpose of this appendix is to provide a more complete version of that part of the appendix of Beat the Dealer. Thorp's tables give expected values for single deck, dealer stands on soft seventeen, player can double down after splitting. My tables give expected values for that, and also six decks, dealer hits soft seventeen, and no doubling down after splitting. Each dealer upcard has a separate page. If the dealer upcard could turn into a soft seventeen, there are separate pages for the two dealer actions on soft seventeen. (What the dealer does on soft seventeen does not

ApPENDIX

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299

matter for upcards of 7, 8, 9, and 10.) The first set of tables is for single deck; and then comes a set of tables for six decks. Each table has· four parts, one each for standing, hitting, doubling down, and splitting. For the two-card hand in which you are interested, find the row corresponding to one of your cards and the column corresponding to the other. (For pair splits you need find only the column for one of your cards.) Here is an example using the tables. Find the table for single deck, dealer stands on soft seventeen and shows a 2. Look at the various ways you can play 4-4 against 2. For standing you should find -.291. That means if you are so foolish as to stand on 4-4 against 2, on average you will lose 29.1 % of your bet. You will not lose it 100% of the time because if the dealer busts you win. For hitting 4-4 against 2 you should fmd -.014. For doubling down on 4-4 against 2 you should find -.186. The value for doubling down is less than twice the value of hitting since if 4-4 is hit with a small enough card you would hit again. The -.186 for doubling down is expressed as a fraction of your initial bet. That is, doubling down 4-4 against 2 costs you 18.6% of your initial bet, which is 9.3% of your doubled bet. The expectation of splitting 4-4 against 2 depends on whether you can double down after splitting; it is -.238 if you are not allowed to double down after splitting, and -.152 if you

are allowed to double after splitting. As with the doubledown numbers, the expectations for splitting are expressed in units of original bets. Thus losing 16.0% of the original bet means losing 8.0% on each of the two split hands. For the example of 4-4 against 2 the highest expected value is the -.014 for hitting, which is why basic strategy recommends hitting 4-4 against 2.

300


To calculate such tables, certain assumptions and approximations must be made, and different assumptions or approximations result in different numbers. The method of calculation I use is to remove three cards maximum for the dealer and three cards maximum for the player (three cards per hand on splits), and then sample with replacement from the remaining pack for any other cards needed to complete the player or dealer hand. Sampling with replacement is the same thing as assuming an infinite number of decks, each of which is missing the cards I have removed. Specifying three dealer cards and three player cards is more accurate than simply assuming each denomination has 1/13 chance of popping up as the next card, but requires considerably more calculation time. Removing a fourth card for the dealer and a fourth card for the player would take about ten times as much calculation time (not 100 times as much because most hands do not require a fourth card), and would result in a small increase in accuracy. Thorp's 1966 numbers are more precise than mine, but only slightly so. The main difference is on splits: Thorp assumes resplits to four hands, and I assume no resplits. If splitting is the best way to play a pair, the tables will say so no matter whether the calculations assume no resplits, unlimited resplits, or something in between. If splitting is recommended, then basic strategy will also say to resplit. Therefore, the tables yield good advice on splits and resplits no matter how the calculations on resplits are done. Another difference between Thorp's numbers and mine is I use total-dependent basic strategy to complete the player hands, and Thorp uses a mixture of totaldependent and composition-dependent. For example,

ApPENDIX

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301

against a dealer's 3, Thorp uses a hard standing nwnber of twelve for analyzing hands starting with 7-2, but a hard standing number of thirteen for hands starting with 6-2, 6-3, 7-3, 8-2, or 8-3. That means he stands on 72-3 if the 3 is the hit card, but hits if 2 is the hit card. I think that standing on 7-2-3 but hitting 7-3-2 is silly; I hit all twelves against a 3. If you compare this appendix with other parts of this book, you might fmd small inconsistencies. Those inconsistencies are unimportant for how fast you can win from casinos. This appendix has been calculated using a more time-conswning but more accurate method than was used for the main part of the book and appendixes A and B. Probabilities for dealer showing an ace or 10 are conditional on the dealer not having a natural. For player naturals, the payoff is 1.5 units in most casinos. However, the tables in this appendix do not show 1.500 for those hands. What the tables show for player hands of A-I0 are the expected values of playing the hand as an ordinary twenty-one. The expected values are less than 1.000 to the extent the dealer also gets to twenty-one in any number of cards. Though these values for A-I0 have no meaning in casinos where naturals win automatically, substituting 1.500 into the tables in those places would have meant eliminating nonobvious numbers that you conceivably might want. If you are fascinated by how the calculations are done and want to delve further into it, I suggest The Theory of Blaclg·ack by Peter A. Griffin.


302


1 Deck, S17, Dealer Shows Ace EV Stand 2

3

4

5

6

7

8

9

-.662 -.673 -.658 -.652 -.646 -.640 -.658 -.659 -.660 -.661

-.662 -.673 -.657 -.651 -.645 -.639 -.657 -.659 -.661 -.660

-.659 -.671 -.655 -.649 -.642 -.637 -.655 -.656 -.659 -.659

-.660 -.670 -.654 -.648 -.641 -.635 -.654 -.655 -.657 -.658

-.483 -.654 -.637 -.631 -.624 -.618 -.635 -.637 -.639 -.640

-.101 -.467 -.644 -.638 -.631 -.624 -.641 -.642 -.645 -.646

.290 -.082 -.452 -.644 -.638 -.631 -.648 -.649 -.651 -.652

.681 .308 -.055 -.452 -.637 -.654 -.655 -.657 -.658

2

3 -.101 -.393 -.375 .171 .087 -.078 -.217 -.345 -.333 -.291

4 -.154 -.445 -.383 -.375 .171 .081 -.070 -.209 -.345 -.334

5

6 -.200 -.509 -.496 -.453 -.444 -.386 .173 .081 -.078 -.226

7

-.206 -.499 -.441 -.433 -.377 .173 .091 -.070 -.217 -.331

-.109 -.556 -.495 -.456 -.495 -.444 -.377 .171 .087 -.083

8 -.017 -.633 -.545 -.495 -.456 -.453 -.433 -.375 .171 .086

.067 -.743 -.625 -.545 -.495 -.496 -.441 -.383 -.375 .170

10 .925 .650 .308 -.082 -.467 -.654 -.670 -.671 -.673 -.673

Ace -.664 .925 .681 .290 -.101 -.483 -.660 -.659 -.662 -.662

10 .152 -.883 -.743 -.633 -.556 -.509 -.499 -.445 -.393 -.349

Ace

-.031 .152 .067 -.017 -.109 -.200 -.206 -.154 -.101 -.068

2 4 3 7 6 5 9 10 8 -.594 -.597 -.621 -.657 -.528 -.358 -.193 -.041 .145 -.811 -.859 -.932 -1.013 -1.018 -1.112 -1.266 -1.485 -1.766 .189 -.859 -.838 -.921 -1.008 -.991 -1.089 -1.250 -1.485 .025 .205 -.847 -.931 -.942 -.928 -.990 -1.089 -1.266 -.403 .040 .222 -.838 -.937 -1.030 -.928 -.991 -1.112 -.800 -.402 .048 .240 -.835 -.937 -.942 -1.008 -1.018 -1.126 -.795 -.396 .055 .240 -.838 -.931 -.921 -1.013 -1.297 -1.127 -.797 -.396 .048 .222 -.847 -.838 -.932 -1.320 -1.299 -1.127 -.795 -.402 .040 .205 -.859 -.859 -1.321 -1.320 -1.297 -1.126 -.800 -.403 .025 .189 -.811

Ace -.600 .145 -.041 -.193 -.358 -.528 -.657 -.621 -.597 -.594

Ace

10 9

8 7 6 5 4 3 2

-.644

EV Hit Ace

10 9 8 7 6 5 4 3 2

-.068 -.349 .170 .086 -.083 -.226 -.331 -.334 -.291 -.258

9

EV Double Down Ace

10 9

8 7 6 5 4 3 2

EV Split reg dbs

2

3

4

5

6

7

8

9

10

Ace

-.428 -.428

-.471 -.471

-.547 -.547

-.648 -.648

-.638 -.638

-.607 -.607

-.340 -.340

-.072 -.072

.098

.224 .224

.098

ApPENDIX

E

303

I Deck, HI7, Dealer Shows Ace EV Stand 2

3

4

S

6

7

8

9

10

Ace

-.590 -.602 -.588 -.582 -.575 -.581 -.586 -.585 -.586 -.588

-.588 -.600 -.586 -.579 -.573 -.579 -.585 -.583 -.585 -.586

-.586 -.598 -.584 -.577 -.570 -.577 -.582 -.581 -.583 -.585

-.588 -.599 -.585 -.578 -.571 -.576 -.584 -.582 -.585 -.586

-.508 -.595 -.580 -.573 -.566 -.571 -.576 -.577 -.579 -.581

-.224 -.505 -.575 -.568 -.561 -.566 -.571 -.570 -.573 -.575

.197 -.212 -.492 -.575 -.568 -.573 -.578 -.577 -.579 -.582

.625 .213 -.187 -.492 -.575 -.580 -.585 -.584 -.586 -.588

.907 .594 .213 -.212 -.505 -.595 -.599 -.598 -.600 -.602

-.592 .907 .625 .197 -.224 -.508 -.588 -.586 -.588 -.590

2

3

4

S

6

7

8

9

10

Ace

-.108 -.380 .130 .038 -.142 -.279 -.366 -.363 -.323 -.294

-.140 -.421 -.405 .133 .039 -.125 -.280 -.377 -.363 -.323

-.189 -.469 -.411 -.404 .136 .044 -.125 -.268 -.377 -.363

-.243 -.524 -.468 -.461 -.407 .141 .042 -.125 -.280 -.366

-.230 -.529 -.518 -.477 -.467 -.405 .141 .044 -.125 -.279

-.174 -.577 -.521 -.483 -.524 -.467 -.407 .136 .039 -.142

-.077 -.645 -.566 -.520 -.483 -.477 -.461 -.404 .133 .038

.018 -.748 -.637 -.566 -.521 -.518 -.468 -.411 -.405 .130

.112 -.884 -.748 -.645 -.577 -.529 -.524 -.469 -.421 -.380

-.071 .112 .018 -.077 -.174 -.230 -.243 -.189 -.140 -.108

2 3 4 5 6 7 8 9 10 -.553 -.550 -.571 -.609 -.531 -.416 -.233 -.064 .136 -.818 -.877 -.952 -1.052 -1.058 -1.154 -1.290 -1.495 -1.769 .181 -.865 -.851 -.948 -1.043 -1.042 -1.132 -1.274 -1.495 .002 .202 -.850 -.944 -.966 -.972 -1.040 -1.132 -1.290 -.434 .017 .223 -.839 -.948 -1.060 -.972 -1.042 -1.154 -.819 -.424 .030 .233 -.833 -.948 -.966 -1.043 -1.058 -1.075 -.815 -.420 .026 .233 -.839 -.944 -.948 -1.052 -1.160 -1.069 -.810 -.420 .030 .223 -.850 -.851 -.952 -1.172 -1.160 -1.069 -.815 -.424 .017 .202 -.865 -.877 -1.175 -1.172 -1.160 -1.075 -.819 -.434 .002 .181 -.818

Ace

Ace

10 9 8 7 6 5 4 3 2

EV Hit Ace

10 9 8 7 6 5 4 3 2

EV Double Down Ace

10 9 8 7 6

5 4

3 2

-.560 .136 -.064 -.223 -.416 -.531 -.609 -.571 -.550 -.553

EV Split reg dbs

2

3

4

-.502

-.542 -.528

-.605 -.587

-.491

5 -.716 -.697

6

7

8

9

10

Ace

-.684 -.664

-.687 -.669

-.484 -.471

-.193 -.186

-.004 -.004

.214 .214

304


1 Deck, S17, Dealer Shows 2 EV Stand Ace

10 9 8 7 6 5 4

3 2

2

3

4

S

6

7

8

9

-.284 -.312 -.288 -.283 -.282 -.278 -.275 -.292 -.294 -.292

-.285 -.314 -.289 -.285 -.282 -.280 -.277 -.294 -.296 -.294

-.285 -.312 -.287 -.283 -.280 -.277 -.275 -.291 -294 -.292

-.268 -.296 -.271 -.266 -.264 -.260 -.257 -.275 -.277 -.275

-.133 -.299 -.275 -.269 -.267 -.263 -.260 -.277 -.280 -.278

.135 -.159 -.278 -.274 -.270 -.267 -.264 -.280 -.282 -.282

.401 .118 -.138 -.276 -.274 -.269 -.266 -.283 -.285 -.283

.656 .385 .137 -.138 -.278 -.275 -.271 -.287 -.289 -.288

2

3

4

5

6

7

8

.039 -.244 .263 .212 .082 -.014 -.101 -.152 -.133 -.115

.016 -.305 -.267 .267 .214 .091 -.018 -.125 -.155 -.133

-.013 -.369 -.293 -.260 .272 .217 .092 -.014 -.125 -.152

-.033 -.436 -.359 -.332 -.253 .283 .223 .092 -.018 -.101

.006 -.466 -.430 -.363 -.332 -.253 .283 .217 .091 -.014

.064 -.539 -.456 -.391 -.407 -.332 -.253 .272 .214 .082

.119 -.633 -.530 -.454 -.391 -.363 -.332 -.260 .267 .212

10 .880 .627 .385 .118 -.159 -.299 -.296 -.312 -.314 -.312

-.276 .880 .656 .401 .135 -.133 -.268 -.285 -.285 -.284

9

10

Ace

.190 -.750 -.628 -.530 -.456 -.430 -.359 -.293 -.267 .263

.246 -.847 -.750 -.633 -.539 -.466 -.436 -.369 -.305 -.244

.094 .246 .190 .119 .064 .006 -.033 -.013 .016 .039

10 .486

-.021

-.931 -tOn -1.265 -1.499 -1.693

.486

-.861 -.726 -.665 -.506 .566 .431 .110 -.165

.378 .236 .126 .011 -.084 -.072 -.049 -.044

Ace

EV Hit Ace

10 9

8 7 6 5 4 3 2

EV Double Down 2 Ace

10 9 8 7

6 5 4

3 2

-.044 -.488 .524 .423 .116 -.165 -.404 -.563 -.589 -.584

3 -.049 -.609 -.534 .532 .425 .110 -.170 -.431 -.570 -.589

4

5

6

7

8

9

-.072 -.738 -.587 -.521 .541 .431 .112 -.186 -.431 -.563

-.084 -.873 -.718 -.663 -.506 .566 .445 .112 -.170 -.404

.011

.126

.236

.378

-.912 -1.060 -1.255 -1.499 -.781 -.908 -1.060 -1.265 -.813 -.781 -.912 -1.077 -.665 -.726 -.861 -.931 -.506 -.663 -.718 -.873 .541 -.521 -.587 -.738 .425 .532 -.534 -.609 .116 .423 .524 -.488

Ace

EV Split reg dbs

2

3

4

5

6

7

8

9

10

Ace

-.130 -.041

-.204 -.120

-.238 -.152

-.234 -.160

-.222 -.129

-.165 -.075

.015 .087

.168 .201

.309 .309

.565 .565

E

ApPENDIX

305

1 Deck, H17, Dealer Shows 2 EV Stand Ace 10 9 8 7 6

S 4 3

2

2

3

4

5

6

7

8

9

10

Ace

-.279 -.305 -.281 -.276 -.274 -.271 -.268 -.286 -.287 -.285

-.280 -.306 -.282 -.278 -.275 -.273 -.270 -.288 -.289 -.287

-.280 -.306 -.282 -.277 -.274 -.271 -.269 -.287 -.288 -.286

-.262 -.289 -.264 -.259 -.256 -.253 -.250 -.269 -.270 -.268

-.135 -.292 -.268 -.262 -.260 -.256 -.253 -.271 -.273 -.271

.127 -.162 -.271 -.267 -.263 -.260 -.256 -.274 -.275 -.274

.395 .107 -.141 -.270 -.267 -.262 -.259 -.278 -.276

.652 .376 .125 -.141 -.271 -.268 -.264 -.282 -.282 -.281

.878 .622 .376 .107 -.162 -.292 -.289 -.306 -.306 -.305

-.272 .878 .652 .395 .127 -.135 -.262 -.280 -.280 -.279

2

3 .018 -.305 -.267 .267 .212 .089 -.020 -.124 -.150 -.129

4

5 -.031 -.438 -.360 -.332 -.253 .282 .221 .091 -.020 -.100

6

7

.062 -.540 -.458 -.392 -.408 -.333 -.253 .271 .212 .080

9 .189 -.750 -.628 -.532 -.458 -.432 -.360 -.294 -.267 .262

Ace

.006 -.467 -.432 -.364 -.333 -.253 .282 .216 .089 -.017

8 .117 -.634 -.532 -.456 -.392 -.364 -,332 -.261 .267 .211

10

-.011 -.370 -.294 -.261 .271 .216 .091 -.015 -.124 -.149

.246 -.847 -.750 -.634 -.540 -.467 -.438 -.370 -.305 -.244

.093 .246 .189 .117 .062 .006 -.031 -.011 .018 .040

-.2n

EV Hit Ace 10 9 8 7 6 5

4 3 2

.040 -.244 .262 .211 .080 -.017 -.100 -.149 -.129 -.112

EV Double Down Ace 10 9 8 7 6 5 4 3 2

2

3

4

5

-.041 -.489 .523 .422 .114 -.166 -.398 -.552 -.574 -.569

-.045 -.611 -.534 .532 .423 .108 -.171 -.426 -.556 -.574

-.068 -.740 -.588 -.521 .541 .430 .111 -.187 -.426 -.552

-.079 -.876 -.721 -.665 -.507 .565 .443 .111 -.171 -.398

7 9 10 6 8 .012 .123 .233 .377 .485 -.935 -1.080 -1.267 -1.500 -1.694 -.864 -.916 -1.064 -1.257 -1.500 -.728 -.785 -.912 -1.064 -1.267 -.666 -.816 -.785 -.916 -1.080 -.506 -.666 -.728 -.864 -.935 .565 -.507 -.665 -.721 -.876 .430 .541 -.521 -.588 -.740 .108 .423 .532 -.534 -.611 -.166 .114 .422 .523 -.489

3 -.199 -.114

4

5 -.228 -.153

6

7

-.234 -.148

-.214 -.122

-.165 -.076

Ace -.019 .485

.3n

.233 .123 .012 -.079 -.068 -.045 -.041

EV Split 2

reg dbs

-.130 -.041

8 .009 .081

9 .164 .197

10 .306 .306

Ace .564 .564

306


1 Deck, S17, Dealer Shows 3 EV Stand Ace 10 9 8

7 6 5 4 3 2

2

3

4

S

6

7

8

9

10

Ace

-.242 -.269 -.265 -.239 -.235 -.232 -.230 -.233 -.249 -.252

-.241 -.267 -.263 -.237 -.233 -.229 -.228 -.231 -.247 -.249

-.225 -.252 -.247 -.221 -.217 -.213 -.211 -.215 -.231 -.233

-.222 -.249 -.245 -.218 -.214 -.210 -.207 -.211 -.228 -.230

-.094 -.251 -.247 -.221 -.216 -.212 -.210 -.213 -.229 -.232

.166 -.120 -.251 -.225 -.221 -.216 -.214 -.217 -.233 -.235

.420 .144 -.122 -.230 -.225 -.221 -.218 -.221 -.237 -.239

.644 .384 .122 -.122 -.251 -.247 -.245 -.247 -.263 -.265

.884 .636 .384 .144 -.120 -.251 -.249 -.252 -.267 -.269

-.233 .884 .644 .420 .166 -.094 -.222 -.225 -.241 -.242

EV Hit Ace 10 9 8

7 6 5 4 3 2

2

3

4

5

6

7

8

9

10

Ace

.071 -.220 .269 .241 .130 .022 -.063 -.108 -.100 -.083

.044 -.284 -.256 .294 .246 .138 .018 -.084 -.120 -.100

.022 -.356 -.281 -.230 .304 .249 .139 .027 -.084 -.108

-.004 -.430 -.356 -.306 -.222 .313 .253 .139 .018 -.063

.035 -.461 -.432 -.347 -.304 -.223 .313 .249 .138 .022

.093 -.536 -.459 -.381 -.389 -.304 -.222 .304 .246 .130

.172 -.634 -.536 -.450 -.381 -.347 -.306 -.230 .294 .241

.196 -.713 -.638 -.536 -.459 -.432 -.356 -.281 -.256 .269

.270 -.846 -.713 -.634 -.536 -.461 -.430 -.356 -.284 -.220

.120 .270 .196 .172 .093 .035 -.004 .022 .044 .071

6 7 8 9 10 .072 .188 .345 .391 .535 -.923 -1.073 -1.267 -1.425 -1.693 -.865 -.918 -1.071 -1.277 -1.425 -.694 -.763 -.900 -1.071 -1.267 -.609 -.n7 -.763 -.918 -1.073 -.445 -.609 -.694 -.865 -.923 .628 -.445 -.612 -.713 -.860 .500 .610 -.460 -.561 -.712 .188 .491 .589 -.512 -.567 -.080 .193 .482 .537 -.439

Ace

EV Double Down Ace 10

9 8 7 6

5 4

3 2

2

3

4

S

.027 -.439 .537 .482 .193 -.080 -.313 -.448 -.499 -.503

.009 -.567 -.512 .589 .491 .188 -.085 -.326 -.474 -.499

.001 -.712 -.561 -.460 .610 .500 .194 -.084 -.326 -.448

-.021 -.860 -.713 -.612 -.445 .628 .509 .194 -.085 -.313

.053 .535 .391 .345 .188 .072 -.021 .001 .009 .027

EV Split reg dbs

2

3

4

5

6

7

8

9

10

Ace

-.072 .015

-.129 -.038

-.129 -.033

-.153 -.075

-.137 -.038

-.086 .000

.075 .155

.169 .210

.363 .363

.612 .612

ApPENDIX

E

307

1 Deck, H17, Dealer Shows 3 EV Stand 2 Ace

10 9 8 7 6 S 4

3 2

-.238 -.264 -.260 -.234 -.230 -.227 -.225 -.228 -.246 -.246

3 -.238 -.263 -.259 -.233 -.229 -.225 -.224 -.227 -.245 -.246

2

3

4

S

6

7

.071 -.220 .268 .240 .129 .020 -.063 -.105 -.098 -.081

.044 -.284 -.256 .294 .245 .137 .016 -.083 -.118 -.098

.023 -.357 -.281 -.230 .304 .248 .138 .026 -.083 -.105

-.003 -.431 -.357 -.306 -.223 .312 .252 .138 .016 -.063

.035 -.463 -.434 -.348 -.305 -.223 .312 .248 .137 .020

4

S

6

.005 -.713 -.563 -.461 .610 .499 .193 -.084 -.322 -.437

-.018 -.862 -.714 -.613 -.445 .628 .507 .193 -.085 -.308

4

5

6

7

-.221 -.246 -.242 -.216 -.211 -.208 -.205 -.209 -.227 -.228

-.218 -.243 -.240 -.213 -.208 -.204 -.202 -.205 -.224 -.225

-.095 -.246 -.242 -.216 -.211 -.207 -.204 -.208 -.225 -.227

.160 -.122 -.246 -.220 -.216 -.211 -.208 -.211 -.229 -.230

8 .415 .135 -.124 -.225 -.220 -.216 -.213 -.216 -.233 -.234

.641 .377 .114 -.124 -.246 -.242 -.240 -.242 -.259 -.260

.091 -.538 -.461 -.383 -.389 -.305 -.223 .304 .245 .129

8 .171 -.634 -.537 -.451 -.383 -.348 -.306 -.230 .294 .240

.195 -.713 -.639 -.537 -.461 -.434 -.357 -.281 -.256 .268

7

8

9

9

10 .883 .632 .377 .135 -.122 -.246 -.243 -.246 -.263 -.264

Ace

-.231 .883 .641 .415 .160 -.095 -.218 -.221 -.238 -.238

EV Hit Ace

10 9 8 7 6 S 4 3 2

9

10 .270 -.846 -.713 -.634 -.538 -.463 -.431 -.357 -.284 -.220

Ace

.119 .270 .195 .171 .091 .035 -.003 .023 .044 .071

EV Double Down 2

Ace

10 9 8 7 6 S

4

3 2

.029 -.440 .537 .480 .192 -.081 -.308 -.437 -.491 -.493

3 .011 -.568 -.513 .589 .490 .187 -.085 -.322 -.470 -.491

.073 .185 .343 .390 -.926 -1.075 -1.269 -1.426 -.867 -.921 -1.074 -1.278 -.696 -.765 -.903 -1.074 -.610 -.779 -.765 -.921 -.445 -.610 -.696 -.867 .628 -.445 -.613 -.714 .499 .610 -.461 -.563 .187 .490 .589 -.513 -.081 .192 .480 .537

10 .535 -1.693 -1.426 -1.269 -1.075 -.926 -.862 -.713 -.568 -.440

Ace

.055 .535 .390 .343 .185 .073 -.018 .005 .011 .029

EV Split reg dbs

2

3

4

S

6

7

-.071 .015

-.128 -.036

-.124 -.027

-.148 -.070

-.132 -.032

-.086 .000

8 .071 .151

9

.166 .208

10 .361 .361

Ace

.612 .612

308



10 9

8 7 6 5 4 3 2

2

3

4

5

6

7

8

9

10

Ace

-.187 -.213 -.208 -.203 -.178 -.172 -.170 -.173 -.177 -.193

-.171 -.198 -.192 -.188 -.162 -.156 -.154 -.157 -.160 -.177

-.167 -.194 -.189 -.184 -.158 -.153 -.150 -.153 -.157 -.173

-.164 -.192 -.187 -.182 -.156 -.150 -.147 -.150 -.154 -.170

-.037 -.194 -.190 -.185 -.159 -.153 -.150 -.153 -.156 -.172

.204 -.065 -.195 -.191 -.165 -.159 -.156 -.158 -.162 -.178

.415 .164 -.085 -.216 -.191 -.185 -.182 -.184 -.188 -.203

.654 .404 .167 -.085 -.195 -.190 -.187 -.189 -.192 -.208

.884 .645 .404 .164 -.065 -.194 -.192 -.194 -.198 -.213

-.179 .884 .654 .415 .204 -.037 -.164 -.167 -.171 -.187

10 .297 -.846 -.712 -.597 -.535 -.455 -.418 -.337 -.258 -.194

Ace

9 10 8 .153 .312 .372 .458 .594 -.909 -1.070 -1.195 -1.423 -1.693 -.840 -.906 -1.082 -1.194 -1.423 -.676 -.760 -.922 -1.082 -1.195 -.555 -.738 -.760 -.906 -1.070 -.381 -.555 -.676 -.840 -.909 .703 -.384 -.584 -.671 -.836 .585 .684 -.429 -.509 -.673 .289 .570 .624 -.452 -.516 .022 .271 .504 .599 -.389

Ace

EV Hit Ace

10 9

8 7 6 S 4 3 2

2

3

4

5

6

7

8

9

.110 -.194 .300 .252 .167 .080 -.011 -.056 -.042 -.036

.091 -.258 -.226 .312 .285 .185 .086 -.018 -.048 -.042

.061 -.337 -.255 -.214 .342 .292 .189 .097 -.018 -.056

.037 -.418 -.336 -.292 -.192 .352 .295 .189 .086 -.011

.077 -.455 -.420 -.338 -.277 -.190 .352 .292 .185 .080

.156 -.535 -.453 -.380 -.369 -.277 -.192 .342 .285 .167

.186 -.597 -.541 -.461 -.380 -.338 -.292 -.214 .312 .252

.229 -.712 -.597 -.541 -.453 -.420 -.336 -.255 -.226 .300

.142 .297 .229 .186 .156 .077 .037 .061 .091 .110

EV Double Down Ace

10 9 8 7 6 5 4

3 2

2

3

4

5

.114 -.389 .599 .504 .271 .022 -.196 -.325 -.353 -.386

.108 -.516 -.452 .624 .570 .289 .040 -.179 -.304 -.353

.084 -.673 -.509 -.429 .684 .585 .296 .043 -.179 -.325

.061 -.836 -.671 -.584 -.384 .703 .589 .296 .040 -.196

2

3

4

5

6

7

8

9

10

Ace

-.009 .105

.008 .134

-.015 .121

-.040 .068

-.029 .110

.015 .153

.104 .194

.252 .298

.416 .416

.668 .668

6

7

.135 .594 .458 .372 .312 .153 .061 .084 .108 .114

EV Split reg dbs

ApPENDIX

E

309

1 Deck, H17, Dealer Shows 4 EV Stand Ace

10 9 8 7 6 5 4

3 2

2

3

4

5

6

7

8

9

10

Ace

-.183 -.208 -.203 -.199 -.173 -.167 -.165 -.168 -.171 -.189

-.166 -.191 -.186 -.181 -.155 -.150 -.147 -.150 -.154 -.171

-.162 -.188 -.183 -.177 -.152 -.146 -.143 -.146 -.150 -.168

-.160 -.185 -.181 -.176 -.149 -.144 -.141 -.143 -.147 -.165

-.039 -.188 -.184 -.179 -.153 -.147 -.144 -.146 -.150 -.167

.196 -.068 -.189 -.185 -.159 -.153 -.149 -.152 -.155 -.173

.410 .154 -.088 -.210 -.185 -.179 -.176 -.177 -.181 -.199

.651 .396 .156 -.088 -.189 -.184 -.181 -.183 -.186 -.203

.883 .640 .396 .154 -.068 -.188 -.185 -.188 -.191 -.208

-.176 .883 .651 .410 .196 -.039 -.160 -.162 -.166 -.183

EV Hit Ace

10 9

8 7 6 S 4 3 2

2

3

4

5

6

7

8

9

10

Ace

.111 -.195 .299 .252 .166 .079 -.010 -.053 -.039 -.034

.092 -.259 -.226 .312 .284 .184 .084 -.016 -.044 -.039

.063 -.337 -.255 -.215 .342 .292 .188 .096 -.016 -.053

.040 -.420 -.337 -.293 -.192 .351 .294 .188 .084 -.010

.077 -.456 -.421 -.339 -.278 -.190 .351 .292 .184 .079

.154 -.537 -.455 -.382 -.370 -.278 -.192 .342 .284 .166

.185 -.598 -.543 -.463 -.382 -.339 -.293 -.215 .312 .252

.229 -.712 -.598 -.543 -.455 -.421 -.337 -.255 -.226 .299

.297 -.846 -.712 -.598 -.537 -.456 -.420 -.337 -.259 -.195

.143 .297 .229 .185 .154 .077 .040 .063 .092 .111

6 7 8 9 10 .154 .309 .370 .457 .594 -.913 -1.073 -1.196 -1.424 -1.693 -.842 -.910 -1.085 -1.196 -1.424 -.678 -.763 -.926 -1.085 -1.196 -.556 -.740 -.763 -.910 -1.073 -.381 -.556 -.678 -.842 -.913 .703 -.385 -.585 -.673 -.839 .583 .685 -.429 -.510 -.675 .287 .568 .624 -.452 -.518 .021 .269 .503 .599 -.389

Ace

EV Double Down Ace

10 9

8 7 6 5 4 3 2

2

3

4

5

.116 -.389 .599 .503 .269 .021 -.192 -.316 -.343 -.379

.112 -.518 -.452 .624 .568 .287 .039 -.173 -.291 -.343

.088 -.675 -.510 -.429 .685 .583 .295 .043 -.173 -.316

.065 -.839 -.673 -.585 -.385 .703 .587 .295 .039 -.192

.137 .594 .457 .370 .309 .154 .065 .088 .112 .116

EV Split reg dbs

2

3

4

5

6

7

8

9

-.006 .108

.015 .141

-.007 .129

-.032 .076

-.022 .117

.016 .155

.100 .190

.249 .296

10 .415 .415

Ace

.668 .668

310



10 9 8 7 6 5 4

3 2

2

3

4

5

6

7

8

9

-.119 -.145 -.139 -.136 -.131 -.104 -.099 -.102 -.105 -.108

-.117 -.142 -.138 -.133 -.128 -.102 -.096 -.099 -.102 -.105

-.114 -.139 -.135 -.132 -.125 -.099 -.093 -.096 -.099 -.102

-.111 -.136 -.132 -.129 -.124 -.096 -.091 -.093 -.096 -.099

.004 -.142 -.138 -.135 -.130 -.103 -.096 -.099 -.102 -.104

.222 -.044 -.164 -.161 -.156 -.130 -.124 -.125 -.128 -.131

.461 .202 -.045 -.166 -.161 -.135 -.129 -.132 -.133 -.136

.682 .448 .203 -.045 -.164 -.138 -.132 -.135 -.138 -.139

10 .893 .674 .448 .202 -.044 -.142 -.136 -.139 -.142 -.145

Ace

-.131 .893 .682 .461 .222 .004 -.111 -.114 -.117 -.119

EV Hit Ace

10 9 8 7 6 5 4

3 2

2

3

4

5

6

7

.159 -.164 .340 .296 .195 .130 .056 .008 .021 .035

.136 -.236 -.197 .349 .307 .234 .140 .048 .008 .021

.108 -.315 -.232 -.187 .361 .339 .246 .153 .048 .008

.082 -.400 -.314 -.270 -.178 .393 .347 .246 .140 .056

.140 -.444 -.402 -.317 -.269 -.163 .393 .339 .234 .130

.174 -.492 -.452 -.373 -.371 -.269 -.178 .361 .307 .195

8 .226 -.591 -.493 -.453 -.373 -.317 -.270 -.187 .349 .296

.268 -.709 -.590 -.493 -.452 -.402 -.314 -.232 -.197 .340

7

8

9

9

10 .331 -.846 -.709 -.591 -.492 -.444 -.400 -.315 -.236 -.164

Ace

10 .662 -1.691 -1.417 -1.182 -.985 -.889 -.799 -.630 -.471 -.328

Ace

10 .460 .460

Ace

.182 .331 .268 .226 .174 .140 .082 .108 .136 .159

EV Double Down Ace

10 9 8 7 6 5 4

3 2

2

3

4

5

6

.211 -.328 .679 .591 .331 .130 -.072 -.187 -.210 -.215

.203 -.471 -.394 .698 .614 .391 .154 -.063 -.185 -.210

.174 -.630 -.465 -.374 .723 .677 .414 .161 -.063 -.187

.147 -.799 -.628 -.356 .787 .694 .414 .154 -.072

.279 -.889 -.804 -.635 -.539 -.325 .787 .677 .391 .130

2

3

4

5

6

7

.127 .272

.116 .270

.094 .252

.067 .196

.066 .236

.038 .192

-.541

.348 .452 .535 -.985 -1.182 -1.417 -.904 -.985 -1.181 -.747 -.906 -.985 -.741 -.747 -.904 -.539 -.635 -.804 -.356 -.541 -.628 .723 -.374 -.465 .614 .698 -.394 .331 .591 .679

.215 .662 .535 .452 .348 .279 .147 .174 .203 .211

EV Split reg dbs

8 .187 .290

9

.338 .391

.732 .732

ApPENDIX

E

311


10 9 8 7 6 S 4

3 2

2

3

4

7

8

9

-.115 -.140 -.135 -.131 -.126 -.099 -.094 -.096 -.100 -.102

-.113 -.137 -.132 -.129 -.123 -.096 -.091 -.093 -.096 -.100

5 -.110 -.134 -.130 -.127 -.122 -.094 -.089 -.091 -.094 -.097

6

-.118 -.142 -.137 -.134 -.129 -.102 -.097 -.100 -.102 -.105

.004 -.140 -.136 -.132 -.127 -.101 -.094 -.096 -.099 -.102

.220 -.045 -.162 -.159 -.154 -.127 -.122 -.123 -.126 -.129

.459 .198 -.046 -.164 -.159 -.132 -.127 -.129 -.131 -.134

.681 .445 .199 -.046 -.162 -.136 -.130 -.132 -.135 -.137

2

3

4

5

6

7

8

9

.159 -.164 .339 .295 .194 .129 .056 .010 .022 .036

.137 -.236 -.197 .349 .307 .234 .140 .049 .009 .022

.108 -.315 -.232 -.187 .361 .338 .245 .153 .049 .010

.082 -.400 -.314 -.270 -.178 .393 .347 .245 .140 .056

.140 -.445 -.403 -.318 -.269 -.163 .393 .338 .234 .129

.174 -.493 -.453 -.374 -.371 -.269 -.178 .361 .307 .194

.226 -.591 -.493 -.454 -.374 -.318 -.270 -.187 .349 .295

.268 -.709 -.591 -.493 -.453 -.403 -.314 -.232 -.197 .339

10 .892 .672 .445 .198 -.045 -.140 -.134 -.137 -.140 -.142

Ace

-.130 .892 .681 .459 .220 .004 -.110 -.113 -.115 -.118

EV Hit Ace

10 9

8 7 6

5 4 3 2

10 .331 -.846 -.709 -.591 -.493 -.445 -.400 -.315 -.236 -.164

Ace

7 8 9 10 .347 .452 .535 .662 -.986 -1.182 -1.418 -1.691 -.906 -.986 -1.181 -1.418 -.748 -.907 -.986 -1.182 -.742 -.748 -.906 -.986

Ace

.182 .331 .268 .226 .174 .140 .082 .108 .137 .159

EV Double Down Ace

10 9 8 7 6 .5 4

3 2

2

3

4

5

6

.212 -.328 .679 .591 .330

.203 -.472 -.394 .698 .614

.175 -.630 -.465 -.374 .723

.148 -.801 -.629 -.541 -.356

.279 -.890 -.806 -.636 -.539

.129

.390

.676

.786

-.325

-.539

-.636

-.070 -.182 -.205 -.210

-.806

-.890

.154 -.060 -.181 -.205

.279

.414 .161 -.060 -.182

.693 .414 .154 -.070

.786 .676 .390 .129

-.356 .723 .614 .330

-.541 -.374 .698 .591

-.629 -.465 -.394 .679

-.801 -.630 -.472 -.328

.148 .175 .203 .212

8 .185 .288

9

10 .495 .495

Ace

.337 .390

.215 .662 .535 .452 .347

EV Split reg dbs

2

3

4

5

6

7

.129 .274

.118 .272

.096 .255

.069 .199

.069 .238

.038 .192

.732 .732

312


1 Deck, S17, Dealer Shows 6 EV Stand 2 Ace

10 9 8 7 6 S 4

3 2

-.115 -.161 -.157 -.152 -.149 -.145 -.118 -.120 -.122 -.125

3 -.112 -.158 -.154 -.150 -.147 -.142 -.116 -.117 -.119 -.122

4 -.110 -.156 -.152 -.148 -.145 -.140 -.113 -.115 -.117 -.120

5

6

7

-.108 -.155 -.151 -.147 -.144 -.140 -.112 -.113 -.116 -.118

.010 -.180

-.173 -.170 -.166 -.140 -.140 -.142 -.145

.262 -.012 -.180 -.178 -.175 -.170 -.144 -.145 -.147 -.149

8 .482 .268 -.012 -.179 -.178 -.173 -.147 -.148 -.150 -.152

6

7

8

9

.133 -.396 -.401 -.323 -.281 -.194 .380 .331 .231 .132

.192 -.483 -.397 -.362 -.367 -.281 -.179 .366 .318

.241 -.586 -.484 -.397 -.362 -.323 -.268 -.184 .356 .307

.280 -.707 -.587 -.484 -.397 -.401 -.309 -.228 -.194 .346

-.1n

9 .694 .484 .265 -.012 -.180

-.1n -.151 -.152 -.154 -.157

10 .899 .697 .484 .268 -.012 -.180 -.155 -.156 -.158 -.161

Ace

-.104 .899 .694 .482 .262 .010 -.108 -.110 -.112 -.115

EV Hit Ace

10 9

8 7 6 S 4

3 2

2

3

4

.169 -.160 .346 .307 .209 .132 .069 .013 .019 .031

.147 -.229 -.194 .356 .318 .231 .163 .059 .013 .019

.120 -.308 -.228 -.184 .366 .331 .263 .175 .059 .013

5 .116 -.387 -.309 -.268 -.179 .380 .362 .263 .163 .069

10 .341 -.845 -.707 -.586 -.483 -.396 -.387 -.308 -.229 -.160

Ace

7 8 9 10 .384 .482 .559 .682 -.967 -1.172 -1.414 -1.690 -.795 -.968 -1.173 -1.414 -.724 -.793 -.968 -1.172 -.734 -.724 -.795 -.967 -.562 -.647 -.802 -.793 -.358 -.536 -.618 -.n5 .731 -.369 -.457 -.616 .636 .712 -.388 -.458 .357 .615 .691 -.319

Ace

10 .520 .520

Ace

.209

.200 .341 .280 .241 .192 .133 .116 .120 .147 .169

EV Double Down Ace

10 9 8 7 6 S 4

3

2

2

3

4

S

6

.230 -.319 .691 .615 .357 .124 -.062 -.216 -.245 -.250

.221 -.458 -.388 .712 .636 .379 .189 -.058 -.215 -.245

.200 -.616 -.457 -.369 .731 .663 .443 .193 -.058 -.216

.216

.266 -.793 -.802 -.647 -.562 -.388 .761 .663 .379 .124

-.n5

-.618 -.536 -.358 .761 .723 .443 .189 -.062

.247 .682 .559 .482 .384 .266 .216 .200 .221 .230

EV Split reg dbs

2

3

4

S

6

7

.125 .278

.111 .263

.083 .240

.055 .190

-.012 .154

.052 .209

8 .233 .339

9

.359 .413

.758 .758

ApPENDIX

E

313


10 9 8 7

6 S 4

3 2

2

J

4

5

6

7

8

9

10

Ace

-.087 -.124 -.122 -.117 -.114 -.109 -.082 -.082 -.085 -.087

-.084 -.122 -.120 -.115 -.111 -.106 -.079 -.079 -.082 -.085

-.082 -.119 -.117 -.113 -.110 -.104 -.077 -.076 -.079 -.082

-.081 -.120 -.117 -.113 -.110 -.105 -.077

-.002 -.145 -.144 -.140 -.137 -.132 -.105 -.104 -.106 -.109

.215 -.031 -.148 -.145 -.142 -.137 -.110 -.110 -.111 -.114

.447 .203 -.032 -.146 -.145 -.140 -.113 -.113 -.115 -.117

.673 .436 .201 -.032 -.148 -.144 -.117 -.117 -.120 -.122

.892 .668 .436 .203 -.031 -.145 -.120 -.119 -.122 -.124

-.085 .892 .673 .447 .215 -.002 -.081 -.082 -.084 -.087

2

3

4

S

8

.154 -.233 -.195 .355 .312 .221 .151 .064 .036 .040

.129 -.314 -.231 -.185 .366 .326 .253 .164 .064 .036

.124 -.398 -.316 -.271 -.179 .379 .354 .253 .151 .072

6 .131 -.409 -.412 -.331 -.284 -.192 .379 .326 .221 .119

7

.174 -.161 .344 .301 .199 .119 .072 .036 .040 .050

.181 -.494 -.410 -.373 -.375 -.284 -.179 .366 .312 .199

.233 -.592 -.494 -.409 -.373 -.331 -.271 -.185 .355 .301

9 .275 -.710 -.593 -.494 -.410 -.412 -.316 -.231 -.195

5 .232 -.795 -.631 -.542 -.358 .757 .708 .430 .178 -.037

.262 -.818 -.823 -.661 -.568 -.385 .757 .651 .364 .110

-.on

-.079 -.082

EV Hit Ace

10 9 8 7

6 S 4

3 2

10

Ace

.339 -.846 -.710 -.592 -.494 -.409 -.398 -.314 -.233 -.161

.202 .339 .275 .233 .181 .131 .124 .129 .154 .174

7 8 9 10 .361 .466 .550 .679 -.988 -1.185 -1.419 -1.691 -.820 -.989 -1.186 -1.419 -.746 -.819 -.989 -1.185 -.749 -.746 -.820 -.988 -.568 -.661 -.823 -.818 -.358 -.542 -.631 -.795 .732 -.369 -.463 -.627 .624 .710 -.390 -.467 .340 .603 .688 -.322

Ace

.344

EV Double Down Ace

10 9 8 7 6

5 4

3 2

2

3

4

.245 -.322 .688 .603 .340 .110 -.037 -.147 -.169 -.175

.239 -.467 -.390 .710 .624 .364 .178 -.029 -.146 -.169

.218 -.627 -.463 -.369 .732 .651 .430 .185 -.029 -.147

3 .147 .300

4

S

.125 .283

.092 .227

6

.258 .679 .550 .466 .361 .262 .232

.218 .239 .245

EV Split 2

reg dbs

.157 .309

6 .022 .186

7 .051 .209

8

9

.202 .307

.339 .393

10 .507 .507

Ace

.756 .756

314


1 Deck, Dealer Shows 7 EV Stand Ace

to 9 8 7 6 5 4

3 2

2

3

4

5

-.462 -.473 -.494 -.488 -.486 -.482 -.477 -.467 -.469 -.471

-.461 -.471 -.492 -.486 -.485 -.480 -.476 -.465 -.468 -.469

-.458 -.469 -.490 -.484 -.483 -.479 -.473 -.463 -.465 -.467

-.468 -.479 -.500 -.494 -.493 -.489 -.485 -.473 -.476 -.477

2

3 .061 -.270 -.247 .292 .278 .198 .093 -.069 -.163 -.118

4

5

6

.034 -.342 -.274 -.245 .294 .286 .202 .112 -.069 -.162

-.023 -.365 -.348 -.327 -.258 .298 .279 .202 .093 -.066

7

8

Ace

.614 .389 -.123 -.503 -.502 -.499 -.494 -.484 -.486 -.488

9 .773 .610 .401 -.123 -.508 -.503 -.500 -.490 -.492 -.494

10

.412 -.122 -.508 -.502 -.502 -.498 -.493 -.483 -.485 -.486

.925 .765 .610 .389 -.122 -.484 -.479 -.469 -.471 -.473

-.453 .925 .773 .614 .412 -.090 -.468 -.458 -.461 -.462

7 .175 -.452 -.375 -.324 -.389 -.331 -.258 .294 .278 .184

8 .222 -.567 -.449 -.374 -.324 -.349 -.327 -.245 .292 .268

9 .243 -.699 -.566 -.449 -.375 -.363 -.348 -.274 -.247 .289

10 .286 -.843 -.699 -.567 -.452 -.376 -.365 -.342 -.270 -.212

Ace

.060 -.376 -.363 -.349 -.331 -.265 .298 .286 .198 .093

7

8

9

10 .467 -1.686 -1.397 -1.135

6 -.090 -.484 -.503 -.499 -.498 -.494 -.489 -.479 -.480 -.482

EV Hit Ace

10 9

8 7 6 5 4

3 2

.108 -.212 .289 .268 .184 .093 -.066 -.162 -.118 -.090

.159 .286 .243 .222 .175 .060 -.023 .034 .061 .108

EV Double Down Ace

10 9 8 7

6 5 4

3 2

2

3

4

5

6

-.158 -.496 .455 .416 .160 -.149 -.568 -.871 -.938 -.943

-.175 -.583 -.579 .472 .444 .175 -.133 -.543 -.872 -.938

-.141 -.707 -.600 -.564 .487 .475 .190 -.109 -.543 -.871

-.190 -.736 -.728 -.707 -.585 .500 .466 .190 -.133 -.568

.014 -.752 -.745 -.735 -.714 -.599 .500 .475 .175 -.149

2

3

4

5

6

7

8

9

-.053 .006

-.114 -.056

-.222 -.158

-.298 -.244

-.269 -.201

-.122 -.066

.202 .250

.341 .363

.240 .325 .351 -.904 -1.135 -1.397 -.750 -.898 -1.132 -.666 -.747 -.898 -.823 -.666 -.750 -.714 -.735 -.745 -.585 -.707 -.728 .487 -.564 -.600 .444 .472 -.579 .160 .416 .455

-.904 -.752 -.736 -.707 -.583 -.496

Ace

-.137 .467 .351 .325 .240 .014 -.190 -.141 -.175 -.158

EV Split reg dbs

10 .482 .482

Ace

.540 .540

E

ApPENDIX

315

1 Deck, Dealer Shows 8 EV Stand Ace

10 9 8 7 6 5 4 3 2

2

3

4

5

6

7

-.508 -.518 -.514 -.534 -.529 -.525 -.523 -.523 -.513 -.515

-.505 -.515 -.512 -.532 -.526 -.524 -.520 -.520 -.511 -.513

-.515 -.524 -.521 -.541 -.536 -.533 -.531 -.529 -.520 -.522

-.515 -.525 -.521 -.542 -.536 -.534 -.531 -.531 -.520 -.523

-.385 -.527 -.523 -.543 -.539 -.536 -.534 -.533 -.524 -.525

.121 -.394 -.526 -.546 -.540 -.539 -.536 -.536 -.526 -.529

8

.608 .095 -.415 -.552 -.546 -.543 -.542 -.541 -.532 -.534

9

.785 .577 .064 -.415 -.526 -.523 -.521 -.521 -.512 -.514

10 .928 .783 .577 .095 -.394 -.527 -.525 -.524 -.515 -.518

-.500 .928 .785 .608 .121 -.385 -.515 -.515 -.505 -.508

Ace

EV Hit Ace

10 9 8 7 6 5 4 3 2

2

3

4

5

10

Ace

.040 -.274 .216 .207 .108 -.055 -.217 -.233 -.180 -.140

.035 -.328 -.316 .221 .217 .118 -.056 -.227 -.230 -.180

-.035 -.357 -.339 -.319 .222 .208 .109 -.054 -.227 -.233

-.084 -.418 -.370 -.387 -.321 .230 .208 .109 -.056 -.217

-.065 -.425 -.421 -.369 -.394 -.322 .230 .208 .118 -.055

.048 -.474 -.428 -.380 -.408 -.394 -.321 .222 .217 .108

.158 -.565 -.475 -.426 -.380 -.369 -.387 -.319 .221 .207

.172 -.697 -.566 -.475 -.428 -.421 -.370 -.339 -.316 .216

.221 -.843 -.697 -.565 -.474 -.425 -.418 -.357 -.328 -.274

.094 .221 .172 .158 .048 -.065 -.084 -.035 .035 .040

4

5

6

7

8

9

Ace

-.313 -.254 -.314 -.626 -.706 -.739 .328 -.704 -.724 .294 .330 -.719 .007 .327 .340 -.437 .014 .317 -.854 -.428 .000 -1.024 -.849 -.447 -1.026 -1.000 -.849 -1.030 -1.026 -1.024

-.333 -.844 -.762 -.829 -.711 .365 .323 .000 -.428 -.854

-.230 -.850 -.856 -.775 -.843 -.712 .365 .317 .014 -.437

10 .332 -1.685 -1.395 -1.130 -.948 -.850

6

7

8

9


10 9 8 7 6 5 4 3 2

3

-.016 .190 .230 -.948 -1.130 -1.395 -.856 -.951 -1.133 -.777 -.853 -.951 -.858 -.n7 -.856 -.843 -.775 -.856 -.711 -.829 -.762 .340 -.719 -.724 .327 .330 -.704 .007 .294 .328

-.739 -.706 -.626

-.296 .332 .230 .190 -.016 -.230 -.333 -.314 -.254 -.313

-.844

EV Split reg dbs

2

3

4

5

6

7

8

9

10

Ace

-.212 -.176

-.265 -.225

-.342 -.299

-.448 -.412

-.412 -.368

-.423 -.385

-.100 -.074

.179 .195

.354 .354

.406 .406

316


1 Deck, Dealer Shows 9 EV Stand 2

3

4

5

6

7

8

-.517 -.528 -.524 -.520 -.541 -.537 -.532 -.533 -.533 -.524

-.527 -.536 -.534 -.530 -.550 -.546 -.542 -.542 -.543 -.533

-.526 -.536 -.532 -.529 -.550 -.546 -.542 -.543 -.542 -.533

-.526 -.535 -.532 -.528 -.549 -.545 -.541 -.542 -.542 -.532

-.407 -.539 -.536 -.532 -.551 -.549 -.545 -.546 -.546 -.537

-.179 -.416 -.540 -.536 -.556 -.551 -.549 -.550 -.550 -.541

.288 -.196 -.412 -.517 -.536 -.532 -.528 -.529 -.530 -.520

2

3

4

S

6

7

8

Ace

-.013

10

-.344

9 8

.143 .121 -.052 -.208 -.284 -.303 -.261 -.221

-.059 -.359 -.392 .139 .118 -.051 -.218 -.304 -.309 -.261

-.113 -.414 -.365 -.395 .150 .118 -.051 -.206 -.304 -.303

-.166 -.475 -.426 -.413 -.390 .152 .119 -.051 -.218 -.284

-.134 -.479 -.480 -.437 -.419 -.386 .152 .118 -.051 -.208

-.087 -.526 -.482 -.443 -.475 -.419 -.390 .150 .118 -.052

.005 -.593 -.532 -.487 -.443 -.437 -.413 -.395 .139 .121

Ace

10 9 8 7

6 S 4 3 2

9 .766 .264 -.196 -.412 -.540 -.536 -.532 -.532 -.534 -.524

10 .936 .744 .264 -.196 -.416 -.539 -.535 -.536 -.536 -.528

9 .097 -.698 -.595 -.532 -.482 -.480 -.426 -.365 -.392 .143

10

Ace

.149 -.842 -.698 -.593 -.526 -.479 -.475 -.414 -.359 -.344

-.001 .149 .097 .005 -.087 -.134 -.166 -.113 -.059 -.013

7 6 8 9 10 -.345 -.255 -.060 .110 .216 -.959 -1.053 -1.186 -1.396 -1.684 -.967 -.964 -1.064 -1.189 -1.396 -.888 -.894 -.974 -1.064 -1.186 -.869 -.978 -.894 -.964 -1.053 -.817 -.869 -.888 -.967 -.959 .240 -.826 -.848 -.860 -.952 .165 .225 -.834 -.759 -.839 -.290 .154 .215 -.840 -.752 -.692 -.274 .174 .214 -.746

-.421 .216 .110 -.060 -.255 -.345 -.452 -.422 -.394 -.373

Ace

-.511 .936 .766 .288 -.179 -.407 -.526 -.526 -.527 -.517

EV Hit

7

6 S 4 3 2

EV Double Down 10 9 8 7

6 S 4

3 2

3

4

S

-.373 -.394 -.422 -.746 -.752 -.839 .214 -.840 -.759 .174 .215 -.834 -.274 .154 .225 -.692 -.290 .165 -.927 -.703 -.279 -1.046 -.946 -.701 -1.066 -1.066 -.946 -1.047 -1.066 -1.046

-.452 -.952 -.860 -.848 -.826 .240 .174 -.279 -.703 -.927

2 Ace

Ace

EV Split reg dbs

2

3

4

5

6

7

8

9

-.382 -.358

-.417 -.396

-.492 -.468

-.607 -.584

-.569 -.544

-.564 -.545

-.430 -.406

-.112 -.103

10 .184 .184

Ace

.290 .290

ApPENDIX

E

317

1 Deck, Dealer Shows 10 EV Stand 3 -.538 -.539 -.535 -.532 -.528 -.550 -.546 -.547 -.546 -.546

4

5

6

7

8

9

-.539 -.540 -.537 -.533 -.529 -.551 -.547 -.547 -.546 -.547

-.538 -.539 -.536 -.532 -.528 -.551 -.546 -.547 -.547 -.547

-.537 -.539 -.535 -.531 -.527 -.550 -.545 -.546 -.546 -.547

-.418 -.543 -.539 -.535 -.531 -.552 -.550 -.551 -.550 -.551

-.186 -.412 -.518 -.514 -.510 -.531 -.527 -.528 -.528 -.529

.064 -.155 -.391 -.518 -.514 -.535 -.531 -.532 -.532 -.533

.555 .103 -.133 -.391 -.518 -.539 -.535 -.536 -.535 -.537

2

3

4

5

6

7

8

9

-.089 -.347 .102 .031 -.155 -.250 -.325 -.347 -.308 -.275

-.124 -.393 -.390 .106 .030 -.141 -.254 -.337 -.344 -.308

-.171 -.446 -.400 -.388 .105 .035 -.143 -.245 -.337 -.347

-.228 -.502 -.455 -.449 -.389 .111 .034 -.143 -.254 -.325

-.188 -.507 -.510 -.465 -.450 -.387 .111 .035 -.141 -.250

-.140 -.557 -.512 -.475 -.516 -.450 -.389 .105 .030 -.155

-.086 -.623 -.558 -.512 -.475 -.465 -.449 -.388 .106 .031

.008 -.710 -.624 -.558 -.512 -.510 -.455 -.400 -.390 .102

2

10 .958 .583 .103 -.155 -.412 -.543 -.539 -.539 -.539 -.540

-.531 .958 .555 .064 -.186 -.418 -.537 -.538 -.538 -.539

10 .104 -.837 -.710 -.623 -.557 -.507 -.502 -.446 -.393 -.347

-.047 .104 .008 -.086 -.140 -.188 -.228 -.171 -.124 -.089

2 3 4 5 6 7 8 9 10 -.479 -.480 -.496 -.537 -.433 -.321 -.224 -.052 .139 -.739 -.811 -.899 -1.002 -1.014 -1.113 -1.246 -1.420 -1.674 .143 -.821 -.819 -.911 -1.024 -1.024 -1.116 -1.249 -1.420 .006 .162 -.806 -.910 -.936 -.950 -1.024 -1.116 -1.246 -.436 .014 .171 -.791 -.908 -1.035 -.950 -1.024 -1.113 -.745 -.445 .011 .171 -.804 -.908 -.936 -1.024 -1.014 -.955 -.738 -.438 .018 .171 -.791 -.910 -.911 -1.002 -1.073 -.956 -.739 -.438 .011 .171 -.806 -.819 -.899 -1.093 -1.071 -.956 -.738 -.445 .014 .162 -.821 -.811 -1.094 -1.093 -1.073 -.955 -.745 -.436 .006 .143 -.739

-.468 .139 -.052 -.224 -.321 -.433 -.537 -.496 -.480 -.479

Ace

10 9 8 7

6 S 4 3 2

Ace

EV Hit Ace

10 9

8 7

6 5

4 3 2

Ace

EV Double Down Ace

10 9 8 7

6 5

4

3 2

Ace

EV Split reg dbs

2

3

4

5

6

7

8

9

-.461 -.453

-.508 -.497

-.578 -.566

-.684 -.672

-.651 -.640

-.632 -.619

-.463 -.453

-.266 -.260

10 .060 .060

Ace

.194 .194

318


6 Decks, S17, Dealer Shows Ace EV Stand 2

-.666 -.668 -.666 -.665 -.664 -.663 -.665 -.666 -.666 -.666

Ace

10 9 8 7 6

5 4

3 2

3 -.666 -.668 -.665 -.664 -.664 -.663 -.665 -.665 -.666 -.666

4

5

6

7

8

9

-.666 -.668 -.665 -.664 -.663 -.662 -.665 -.665 -.665 -.666

-.666 -.667 -.665 -.664 -.663 -.662 -.664 -.665 -.665 -.665

-.479 -.665 -.662 -.661 -.660 -.660 -.662 -.662 -.663 -.663

-.100 -.476 -.663 -.662 -.661 -.660 -.663 -.663 -.664 -.664

.280 -.097 -.474 -.663 -.662 -.661 -.664 -.664 -.664 -.665

.660 .283 -.093 -.474 -.663 -.662 -.665 -.665 -.665 -.666

10 .923 .655 .283 -.097 -.476 -.665 -.667 -.668 -.668 -.668

-.667 .923 .660 .280 -.100 -.479 -.666 -.666 -.666 -.666

10 .145 -.857 -.728 -.628 -.557 -.516 -.483 -.441 -.396 -.350

-.023 .145 .079 -.008 -.095 -.183 -.172 -.134 -.096 -.060

Ace

EV Hit Ace

10 9 8 7 6 5 4

3 2

2

3

4

5

6

7

-.060 -.350 .147 .082 -.068 -.202 -.314 -.309 -.281 -.254

-.096 -.396 -.354 .147 .082 -.068 -.200 -.316 -.309 -.281

-.134 -.441 -.395 -.355 .147 .081 -.066 -.199 -.316 -.309

-.172 -.483 -.440 -.403 -.355 .148 .083 -.066 -.200 -.314

-.183 -.516 -.483 -.442 -.405 -.357 .148 .081 -.068 -.202

-.095 -.557 -.514 -.476 -.449 -.405 -.355 .147 .082 -.068

8 -.008 -.628 -.555 -.514 -.476 -.442 -.403 -.355 .147 .082

.079 -.728 -.626 -.555 -.514 -.483 -.440 -.395 -.354 .147

5

6

7

8

9

9

Ace


10 9 8 7

6 5 4

3 2

3

4

-.620 -.620 -.624 -.629 -.536 -.362 -.189 -.019 -.826 -.877 -.932 -.988 -1.032 -1.114 -1.255 -1.455 .121 -.834 -.874 -.930 -.987 -1.027 -1.110 -1.252 -.008 .124 -.832 -.888 -.933 -.974 -1.027 -1.110 -.428 -.006 .127 -.830 -.890 -.947 -.974 -1.027 -.809 -.428 -.004 .130 -.831 -.890 -.933 -.987 -1.129 -.808 -.427 -.002 .130 -.830 -.888 -.930 -1.303 -1.130 -.808 -.427 -.004 .127 -.832 "-.874 -1.332 -1.303 -1.130 -.808 -.428 -.006 .124 -.834 -1.332 -1.332 -1.303 -1.129 -.809 -.428 -.008 .121

10 .115 -1.714 -1.455 -1.255 -1.114 -1.032 -.988 -.932 -.877 -.826

-.620 .115 -.019 -.189 -.362 -.536 -.629 -.624 -.620 -.620

Ace

EV Split reg dbs

2

3

4

S

6

7

8

9

10

Ace

-.410 -.410

-.459 -.459

-.513 -.513

-.571 -.571

-.614 -.614

-.620 -.620

-.386 -.386

-.122 -.122

.153 .153

.127 .127

ApPENDIX

E

319

6 Decks, HI7, Dealer Shows Ace EV Stand 2

3

4

5

6

7

8

9

10

Ace

-.597 -.599 -.597 -.596 -.595 -.596 -.596 -.596 -.597 -.597

-.597 -.599 -.597 -.596 -.595 -.595 -.596 -.596 -.596 -.597

-.597 -.599 -.596 -.595 -.594 -.595 -.596 -.596 -.596 -.596

-.597 -.599 -.596 -.595 -.594 -.595 -.596 -.596 -.596 -.596

-.514 -.598 -.596 -.595 -.594 -.595 -.595 -.595 -.595 -.596

-.225 -.514 -.595 -.594 -.593 -.594 -.594 -.594 -.595 -.595

.190 -.223 -.512 -.595 -.594 -.595 -.595 -.595 -.596 -.596

.606 .192 -.220 -.512 -.595 -.596 -.596 -.596 -.597 -.597

.905 .600 .192 -.223 -.514 -.598 -.599 -.599 -.599 -.599

-.598 .905 .606 .190 -.225 -.514 -.597 -.597 -.597 -.597

2

3

4

5

6

7

8

9

-.100 -.383 .107 .034 -.127 -.266 -.352 -.344 -.318 -.293

-.135 -.427 -.387 .107 .034 -.125 -.266 -.353 -.344 -.318

-.172 -.469 -.425 -.387 .108 .035 -.124 -.264 -.353 -.344

-.209 -.510 -.469 -.433 -.388 .109 .034 -.124 -.266 -.352

-.221 -.540 -.509 -.470 -.434 -.387 .109 .035 -.125 -.266

-.161 -.579 -.539 -.503 -.478 -.434 -.388 .108 .034 -.127

-.065 -.640 -.577 -.539 -.503 -.470 -.433 -.387 .107 .034

.030 -.733 -.638 -.577 -.539 -.509 -.469 -.425 -.387 .107

10 .104 -.858 -.733 -.640 -.579 -.540 -.510 -.469 -.427 -.383

Ace -.065 .104 .030 -.065 -.161 -.221 -.209 -.172 -.135 -.100

2 3 4 S 7 6 8 9 10 -.584 -.584 -.587 -.593 -.549 -.418 -.228 -.041 .108 -.834 -.895 -.959 -1.027 -1.080 -1.158 -1.280 -1.466 -1.717 .115 -.842 -.891 -.958 -1.026 -1.077 -1.154 -1.277 -1.466 -.030 .119 -.839 -.906 -.962 -1.014 -1.0n -1.154 -1.280 -.455 -.028 .122 -.837 -.907 -.9n -1.014 -1.077 -1.158 -.832 -.453 -.026 .124 -.837 -.907 -.962 -1.026 -1.080 -1.086 -.831 -.452 -.026 .124 -.837 -.906 -.958 -1.027 -1.180 -1.085 -.830 -.452 -.026 .122 -.839 -.891 -.959 -1.193 -1.180 -1.085 -.831 -.453 -.028 .119 -.842 -.895 -1.194 -1.193 -1.180 -1.086 -.832 -.455 -.030 .115 -.834

Ace -.585 .108 -.041 -.228 -.418 -.549 -.593 -.587 -.584 -.584

Ace 10 9 8 7 6 5 4 3 2

EV Hit Ace 10 9 8 7 6 S 4 3 2

EV Double Down Ace 10 9 8 7 6 5 4 3 2

EV Split reg dbs

2

3

4

S

6

7

8

9

-.492 -.490

-.538 -.536

-.587 -.585

-.645 -.642

-.682 -.679

-.698 -.695

-.520 -.518

-.240 -.239

10 .055 .055

Ace .121 .121

320


6 Decks, S17, Dealer Shows 2 EV Stand 3 -.291 -.296 -.292 -.291 -.291 -.291 -.290 -.292 -.293 -.293

4

5

6

7

8

9

10

Ace

-.291 -.296 -.292 -.292 -.291 -.291 -.290 -.292 -.293 -.293

-.291 -.296 -.292 -.291 -.291 -.290 -.290 -.292 -.292 -.292

-.289 -.293 -.289 -.289 -.288 -.288 -.287 -.290 -.290 -.290

-.150 -.294 -.290 -.289 -.289 -.288 -.288 -.290 -.291 -.291

.124 -.154 -.290 -.290 -.289 -.289 -.288 -.291 -.291 -.291

.389 .121 -.151 -.290 -.290 -.289 -.289 -.291 -.291 -.292

.642 .386 .124 -.151 -.290 -.290 -.289 -.292 -.292 -.292

.882 .638 .386 .121 -.154 -.294 -.293 -.296 -.296 -.296

-.290 .882 .642 .389 .124 -.150 -.289 -.291 -.291 -.291

2

3

4

5

6

7

8

9

Ace

.045 -.252 .242 .187 .076 -.020 -.108 -.142 -.129 -.115

.021 -.307 -.256 .243 .187 .077 -.021 -.111 -.143 -.129

-.002 -.363 -.305 -.254 .244 .188 .078 -.020 -.111 -.142

-.023 -.420 -.362 -.312 -.253 .246 .189 .078 -.021 -.108

.001 -.470 -.419 -.362 -.312 -.253 .246 .188 .077 -.020

.063 -.537 -.469 -.413 -.369 -.312 -.253 .244 .187 .076

.123 -.624 -.535 -.468 -.413 -.362 -.312 -.254 .243 .187

.183 -.732 -.623 -.535 -.469 -.419 -.362 -.305 -.256 .242

10 .240 -.854 -.732 -.624 -.537 -.470 -.420 -.363 -.307 -.252

.084 .240 .183 .123 .063 .001 -.023 -.002 .021 .045

7 6 8 9 10 -.004 .121 .241 .362 .473 -.940 -1.073 -1.248 -1.465 -1.708 -.838 -.937 -1.071 -1.247 -1.465 -.725 -.825 -.937 -1.071 -1.248 -.624 -.739 -.825 -.937 -1.073 -.507 -.624 -.725 -.838 -.940 .486 -.507 -.623 -.723 -.839 .371 .482 -.509 -.611 -.726 .069 .370 .480 -.511 -.614 -.198 .070 .369 .479 -.504

-.063 .473 .362 .241 .121 -.004 -.074 -.071 -.068 -.067

2 Ace

10 9 8 7 6 S 4 3 2

EV Hit Ace

10 9 8 7 6 S 4

3 2

EV Double Down Ace

10 9 8 7

6 S 4 3 2

2

3

4

S

-.067 -.504 .479 .369 .070 -.198 -.431 -.563 -.585 -.586

-.068 -.614 -.511 .480 .370 .069 -.198 -.434 -.564 -.585

-.071 -.726 -.611 -.509 .482 .371 .070 -.201 -.434 -.563

-.074 -.839 -.723 -.623 -.507 .486 .373 .070 -.198 -.431

Ace

EV Split reg dbs

2

3

4

5

6

7

8

9

-.148 -.083

-.201 -.135

-.230 -.164

-.253 -.188

-.273 -.206

-.210 -.144

-.035 .030

.152 .188

10 .356 .356

Ace

.486 .486

ApPENDIX

E

321

6 Decks, H17, Dealer Shows 2 EV Stand -.285 -.290 -.286 -.285 -.285 -.284 -.284 -.286 -.286 -.286

3 -.285 -.290 -.286 -.285 -.285 -.284 -.284 -.286 -.286 -.286

-.285 -.289 -.286 -.285 -.284 -.284 -.284 -.286 -.286 -.286

5 -.283 -.287 -.283 -.282 -.282 -.281 -.281 -.284 -.284 -.284

2

3

4

5

.046 -.252 .241 .185 .074 -.023 -.108 -.140 -.126 -.113

.022 -.308 -.256 .242 .186 .075 -.023 -.111 -.140 -.126

-.001 -.364 -.306 -.255 .243 .187 .076 -.022 -.111 -.140

-.022 -.422 -.363 -.312 -.254 .245 .188 .076 -.023 -.108

2

Ace 10 9 8 7

6 5 4 3 2

4

6 -.153 -.287 -.284 -.283 -.282 -.282 . -.281 -.284 -.284 -.284

.113 -.157 -.284 -.283 -, -.283 ;:.282 -.282 -.284 -.285 -.285

6 .000 -.472 -.421 -.364 -.313 -.254 .245 .187 .075 -.023

.060 -.538 -.471 -.414 -.371 -.313 -.254 .243 .186 .074

7

10 .880 .633 .378 .110 -.157 -.287 -.287 -.289 -.290 -.290

Ace -.284 .880 .638 .381 .113 -.153 -.283 -.285 -.285 -.285

-.282 -.285 -.285 -.285

9 .638 .378 .113 -.154 . -.284 -.284 -.283 -.286 -.286 -.286

8 .121 -.625 -.537 -.471 -.414 -.364 -.312 -.255 .242 .185

9

10

Ace

.182 -.733 -.624 -.537 -.471 -.421 -.363 -.306 -.256 .241

.239 -.854 -.733 -.625 -.538 -.472 -.422 -.364 -.308 -.252

.082 .239 .182 .121 .060 .000 -.022 -.001 .022 .046

Ace

8 .381 .110 -.154 -.284

~1r. -. 83

EV Hit Ace 10 9 8 7

6 5 4 3 2

7

EV Double Down Ace 10 9 8 7

6 5 4

3 2

2

3

4

5

-.064 -.505 .478 .367 .068 -.200 -.427 -.552 -.573 -.573

-.064 -.616 -.512 .480 .368 .067 -.200 -.430 -.552 -.573

-.068 -.729 -.612 -.510 .482 .369 .068 -.202 -.430 -.552

-.070 -.843 -.726 -.625 -.507 .485 .371 .068 -.200 -.427

6 7 8 9 10 -.005 .116 .237 .360 .472 -.945 -.1.077 -1.250 -1.466 -1.708 -.841 -.942 1.075 -1.249 -1.466 -.727 -.829 -.941 -1.075 -1.250 -.625 -.741 -.829 -.942 -1.0n -.507 -.625 -.727 -.841 -.945 .485 -.507 -.625 -.726 -.843 .369 .482 -.510 -.612 -.729 .067 .368 .480 -.512 -.616 -.200 .068 .367 .478 -.505

-.060 .472 .360 .237 .116 -.005 -.070 -.068 -.064 -.064

3 -.197 -.132

4

-.226 -.160

5 -.248 -.183

6 -.267 -.200

Ace .485 .485

EV Split 2

reg dbs

-.149 -.083

7

-.211 -.145

8 -.040 .024

9 .148 .183

10

.353 .353

322


6 Decks, S17, Dealer Shows 3 EV Stand 2 Ace

10 9 8 7 6

5 4 3

2

-.250 -.255 -.254 -.250 -.249 -.249 -.248 -.249 -.251 -.251

3 -.250 -.254 -.254 -.250 -.249 -.249 -.248 -.249 -.251 -.251

4

-.248 -.252 -.251 -.247 -.247 -.246 -.246 -.246 -.249 -.249

5 -.247 -.252 -.251 -.247 -.246 -.246 -.245 -.246 -.248 -.248

6

7

8

9

10

Ace

-.114 -.252 -.251 -.247 -.247 -.246 -.246 -.246 -.249 -.249

.151 -.118 -.252 -.248 -.247 -.247 -.246 -.247 -.249 -.249

.407 .148 -.118 -.249 -.248 -.247 -.247 -.247 -.250 -.250

.649 .401 .145 -.118 -.252 -.251 -.251 -.251 -.254 -.254

.885 .648 .401 .148 -.118 -.252 -.252 -.252 -.254 -.255

-.249 .885 .649 .407 .151 -.114 -.247 -.248 -,250 -,250

5 .007 -.410 -.350 -.294 -.232 .269 .214 .108 .010 -.074

6

7

10

Ace

.091 -.532 -.463 -.402 -.355 -.293 -.232 .267 .212 .106

8 .153 -.622 -.532 -.462 -.402 -.349 -.294 -.233 .266 .212

9

.030 -.463 -.410 -.349 -.293 -.232 .269 .213 .107 .010

.205 -.726 -.623 -.532 -.463 -.410 -.350 -.289 -.237 .262

.262 -.854 -.726 -.622 -.532 -.463 -.410 -.350 -.290 -.231

,106 .262 .205 .153 ,091 .030 ,007 ,028 ,050 .074

6

7

8

9

10

Ace

.058 .179 .303 .407 .521 -.927 -1.065 -1.244 -1.451 -1.707 -.821 -.926 -1.065 -1.246 -1.451 -.697 -.805 -.923 -1.065 -1.244 -.587 -.710 -.805 -.926 -1.065 -.464 -.587 -.697 -.821 -.927 .535 -.464 -.587 -.700 -.820 .424 .532 -.466 -.579 -.700 .132 .422 .529 -.474 -.580 -.127 .132 .421 .521 -,463

.003 .521 .407 .303 .179 .058 -.009 -.006 -.004 -.001

10 ,404 .404

Ace

EV Hit Ace

10 9 8 7 6

5 4 3 2

2

3

4

.074 -.231 .262 .212 .106 .010 -.074 -.107 -.096 -.082

.050 -.290 -.237 .266 .212 .107 .010 -.077 -.109 -.096

.028 -.350 -.289 -.233 .267 .213 .108 .011 -.077 -.107

EV Double Down Ace

10 9 8 7 6

5 4 3 2

2

3

4

5

-.001 -.463 .521 .421 .132 -.127 -.352 -.477 -.502 -.503

-.004 -.580 -.474 .529 .422 .132 -.127 -.353 -.481 -.502

-.006 -.700 -.579 -.466 .532 .424 .133 -.127 -.353 -.477

-.009 -.820 -.700 -.587 -.464 .535 .425 .133 -.127 -.352

2

3

4

5

6

7

8

9

-.095 -.022

-.136 -.062

-.159 -.084

-.184 -.112

-.203 -.128

-.143 -.069

.026 .098

.198 .238

EV Split reg dbs

.533 .533

ApPENDIX

E

323

6 Decks, H17, Dealer Shows 3 EV Stand 2

Ace 10 9 8 7

6 5

4 3 2

-.245 -.249 -.248 -.244 -.244 -.243 -.243 -.243 -.246 -.246

3 -.245 -.249 -.248 -.244 -.244 -.243 -.243 -.243 -.246 -.246

4

5

-.242 -.246 -.246 -.242 -.241 -.241 -.240 -.241 -.243 -.243

-.242 -.246 -.245 -.241 -.241 -.240 -.240 -.240 -.243 -.243

6 -.116 -.247 -.246 -.242 -.241 -.241 -.240 -.241 -.243 -.243

.142 -.121 -.246 -.242 -.242 -.241 -.241 -.241 -.244 -.244

6 .029 -.465 -.412 -.350 -.294 -.232 .268 .212 .105 .008

.088 -.534 -.465 -.404 -.356 -.294 -.232 .267 .211 .104

9

10

.645 .394 .135 -.121 -.246 -.246 -.245 -.246 -.248 -.248

.884 .644 .394 .138 -.121 -.247 -.246 -.246 -.249 -.249

Ace -.244 .884 .645 .400 .142 -.116 -.242 -.242 -.245 -.245

9

10

Ace

.203 -.726 -.624 -.534 -.465 -.412 -.351 -.290 -.237 .261

.261 -.854 -.726 -.623 -.534 -.465 -.411 -.351 -.291 -.232

.104 .261 .203 .151 .088 .029 .008 .029 .050 .074

6 7 8 9 10 .057 .175 .300 .405 .520 -.931 -1.068 -1.246 -1.452 -1.707 -.824 -.930 -1.068 -1.248 -1.452 -.699 -.808 -.927 -1.068 -1.246 -.588 -.713 -.808 -.930 -1.068 -.465 -.588 -.699 -.824 -.931 .535 -.465 -.589 -.702 -.823 .422 .532 -.467 -.580 -.702 .130 .421 .529 -.475 -.581 -.129 .130 .419 .521 -.464

Ace

7

8 .400 .138 -.121 -.243 -.242 -.242 -.241 -.242 -.244 -.244

EV Hit Ace 10 9 8 7

6 5

4 3 2

2

3

4

5

.074 -.232 .261 .210 .104 .008 -.074 -.104 -.093 -.080

.050 -.291 -.237 .265 .211 .105 .008 -.077 -.106 -.093

.029 -.351 -.290 -.233 .267 .212 .106 .009 -.077 -.104

.008 -.411 -.351 -.294 -.232 .268 .212 .106 .008 -.074

7

8 .151 -.623 -.534 -.464 -.404 -.350 -.294 -.233 .265 .210

EV Double Down Ace 10 9 8 7

6 5 4 3 2

2

3

4

5

.002 -.464 .521 .419 .130 -.129 -.348 -.467 -.492 -.492

-.001 -.581 -.475 .529 .421 .130 -.129 -.350 -.472 -.492

-.003 -.702 -.580 -.467 .532 .422 .131 -.129 -.350 -.467

-.006 -.823 -.702 -.589 -.465 .535 .424 .131 -.129 -.348

.005 .520 .405 .300 .175 .057 -.006 -.003 -.001 .002

EV Split reg dbs

2

3

4

5

6

7

8

9

10

Ace

-.095 -.022

-.133 -.059

-.155 -.080

-.180 -.107

-.198 -.123

-.143 -.070

.021 .093

.194 .234

.402 .402

.533 .533

324


6 Decks, S17, Dealer Shows 4 EV Stand Ace

10 9

8 7 6 S 4

3 2

2

3

4

5

6

7

-.207 -.211 -.211 -.210 -.206 -.205 -.205 -.205 -.205 -.208

-.205 -.209 -.208 -.207 -.203 -.203 -.202 -.203 -.203 -.205

-.204 -.209 -.208 -.207 -.203 -.202 -.202 -.202 -.203 -.205

-.204 -.208 -.207 -.207 -.203 -.202 -.201 -.202 -.202 -.205

-.074 -.209 -.208 -.207 -.203 -.202 -.202 -.202 -.203 -.205

.180 -.078 -.209 -.208 -.204 -.203 -.203 -.203 -.203 -.206

2

3

6

7

.082 -.272 -.215 .288 .239 .138 .046 -.039 -.069 -.058

4 .059 -.335 -.271 -.214 .292 .240 .138 .048 -.039 -.070

5

.104 -.210 .286 .234 .135 .045 -.038 -.070 -.058 -.047

.039 -.399 -.335 -.277 -.210 .294 .240 .138 .046 -.038

.062 -.456 -.400 -.335 -.275 -.210 .294 .240 .138 .045

.124 -.528 -.456 -.393 -.341 -.275 -.210 .292 .239 .135

9

10

Ace

.660 .420 .174 -.081 -.209 -.208 -.207 -.208 -.208 -.211

.888 .658 .420 .174 -.078 -.209 -.208 -.209 -.209 -.211

-.206 .888 .660 .422 .180 -.074 -.204 -.205 -.205 -.207

8

9

.230 -.725 -.614 -.529 -.456 -.400 -.335 -.271 -.215 .286

10 .285 -.853 -.725 -.614 -.528 -.456 -.399 -.335 -.272 -.210

Ace

.177 -.614 -.529 -.457 -.393 -.335

8 9 10 .124 .249 .355 .460 .571 -.912 -1.057 -1.228 -1.449 -1.707 -.799 -.912 -1.058 -1.228 -1.449 -.671 -.786 -.914 -1.058 -1.228 -.550 -.681 -.786 -.912 -1.057 -.420 -.550 -.671 -.799 -.912 .587 -.421 -.554 -.670 -.799 .480 .584 -.427 -.542 -.670 .199 .478 .576 -.431 -.543 -.053 .196 .468 .571 -.421

Ace

10 .454 .454

Ace

8 .422 .174 -.081 -.212 -.208 -.207 -.207 -.207 -.207 -.210

EV Hit Ace

10 9

8 7

6 S 4

3 2

-.2n

-.214 .288 .234

.129 .285 .230 .177 .124 .062 .039 .059 .082 .104

EV Double Down Ace

10 9 8

7

6 S 4

3 2

2

3

4

5

.068 -.421 .571 .468 .196 -.053 -.269 -.390 -.411 -.416

.067 -.543 -.431 .576 .478 .199 -.050 -.266 -.387 -.411

.062 -.670 -.542 -.427 .584 .480 .200 -.050 -.266 -.390

.058 -.799 -.670 -.554 -.421 .587 .481 .200 -.050 -.269

4

5

6

7

8

9

-.086 .008

-.110 -.017

-.128 -.025

-.070 .033

.082 .163

.257 .301

6

7

,071 ,571 .460 ,355 ,249 .124 .058 .062 .067 ,068

EV Split 2

reg dbs

-.038 .048

J

-.060 .030

,583 ,583

ApPENDIX

E

325

6 Decks, HI7, Dealer Shows 4 EV Stand Ace 10 9 8 7 6 S 4 J 2

2

J

4

-.202 -.206 -.205 -.205 -.201 -.200 -.199 -.200 -.200 -.203

-.199 -.204 -.203 -.202 -.198 -.197 -.197 -.197 -.198 -.200

-.199 -.203 -.202 -.202 -.198 -.197 -.196 -.197 -.197 -.200

5 -.199 -.203 -.202 -.201 -.197 -.197 -.196 -.196 -.197 -.199

6

7

-.076 -.203 -.203 -.202 -.198 -.197 -.197 -.197 -.197 -.200

.171 -.081 -.203 -.202 -.198 -.198 -.197 -.198 -.198 -.201

5 .041 -.401 -.336 -.278 -.211 .293 .240 .137 .044 -.038

6

7

.062 -.458 -.401 -.337 -.276 -.210 .293 .239 .137 .044

.122 -.530 -.458 -.395 -.342 -.276 -.211 .292 .238 .134

5 .061 -.801 -.672 -.556 -.421 .587 .479 .198 -.051 -.265

10 .887 .654 .413 .164 -.081 -.203 -.203 -.203 -.204 -.206

Ace -.201 .887 .656 .416 .171 -.076 -.199 -.199 -.199 -.202

9

10

Ace

.229 -.725 -.615 -.531 -.458 -.401 -.336 -.272 -.216 .285

.285 -.853 -.725 -.615 -.530 -.458 -.401 -.336 -.272 -.211

.129 .285 .229 .176 .122 .062 .041 .061 .083 .104

6 7 8 9 10 .123 .245 .352 .459 .570 -.916 -1.060 -1.230 -1.450 -1.707 -.802 -.915 -1.062 -1.230 -1.450 -.673 -.789 -.918 -1.062 -1.230 -.551 -.683 -.789 -.915 -1.060 -.421 -.551 -.673 -.802 -.916 .587 -.421 -.556 -.672 -.801 .479 .584 -.428 -.544 -.673 .197 .4n .575 -.432 -.545 -.054 .194 .467 .571 -.421

Ace

8 .416 .164 -.084 -.206 -.202 -.202 -.201 -.202 -.202 -.205

.656 .413 .164 -.084 -.203 -.203 -.202 -.202 -.203 -.205

8 .176 -.615 -.531 -.459 -.395 -.337 -.278 -.214 .288 .233

9

EV Hit Ace 10 9 8 7 6 S 4 J 2

2

J

4

.104 -.211 .285 .233 .134 .044 -.038 -.067 -.056 -.045

.083 -.272 -.216 .288 .238 .137 .044 -.039 -.066 -.056

.061 -.336 -.272 -.214 .292 .239 .137 .046 -.039 -.067

EV Double Down Ace 10 9

8 7 6 S 4 J 2

2

J

4

.070 -.421 .571 .467 .194 -.054 -.265 -.381 -.401 -.406

.069 -.545 -.432 .575 .477 .197 -.051 -.263 -.401

.065 -.673 -.544 -.428 .584 .479 .198 -.051 -.263 -.381

3 -.055 .035

-.080 .014

-.3n

.073 .570 .459 .352 .245 .123 .061 .065 .069 .070

EV Split 2

reg dbs

-.034 .052

4

5 -.105 -.011

6

7

8

9

-.122 -.020

-.070 .034

.079 .159

.254 .298

10 .452 .452

Ace .582 .582

326



10 9 8

7 6 5

4 3 2

2

3

4

5

6

7

8

9

10

Ace

-.160 -.164 -.163 -.162 -.162 -.158 -.157 -.157 -.158 -.158

-.159 -.163 -.163 -.162 -.161 -.157 -.156 -.157 -.157 -.158

-.159 -.163 -.162 -.162 -.161 -.157 -.156 -.156 -.157 -.157

-.158 -.163 -.162 -.161 -.160 -.156 -.156 -.156 -.156 -.157

-.037 -.163 -.163 -.162 -.161 -.157 -.156 -.157 -.157 -.158

.203 -.045 -.167 -.166 -.165 -.161 -.160 -.161 -.161 -.162

.443 .200 -.045 -.167 -.166 -.162 -.161 -.162 -.162 -.162

.672 .441 .200 -.045 -.167 -.163 -.162 -.162 -.163 -.163

.892 .671 .441 .200 -.045 -.163 -.163 -.163 -.163 -.164

-.161 .892 .672 .443 .203 -.037 -.158 -.159 -.159 -.160

2

3 .116 -.254 -.194 .314 .264 .170 .082 .001 -.028 -.017

4

5

6

7

8

9

10

Ace

.094 -.320 -.253 -.192 .316 .269 .172 .084 .001 -.028

.075 -.388 -.320 -.259 -.191 .321 .270 .172 .082 .002

.099 -.449 -.388 -.321 -.259 -.189 .321 .269 .170 .080

.152 -.518 -.450 -.383 -.329 -.259 -.191 .316 .264 .164

.207 -.611 -.518 -.450 -.383 -.321 -.259 -.192 .314 .263

.258 -.723 -.611 -.518 -.450 -.388 -.320 -.253 -.194 .312

.311 -.853 -.723 -.611 -.518 -.449 -.388 -.320 -.254 -.189

.161 .311 .258 .207 .152 .099 .075 .094 .116 .137

6 7 8 9 10 .198 .304 .414 .516 .622 -.898 -1.036 -1.223 -1.447 -1.706 -.777 -.900 -1.036 -1.223 -1.447 -.642 -.767 -.900 -1.036 -1.223 -.519 -.658 -.767 -.900 -1.036 -.377 -.519 -.642 -.777 -.898 .641 -.382 -.519 -.641 -.776 .538 .632 -.385 -.507 -.641 .266 .529 .628 -.388 -.508 .023 .257 .525 .625 -.377

Ace

EV Hit Ace

10 9 8 7

6 5

4 3 2

.137 -.189 .312 .263 .164 .080 .002 -.028 -.017 -.005

EV Double Down Ace

10 9 8 7 6

5 4 3 2

2

3

4

5

.140 -.377 .625 .525 .257 .023 -.183 -.296 -.315 -.316

.138 -.508 -.388 .628 .529 .266 .026 -.181 -.296 -.315

.134 -.641 -.507 -.385 .632 .538 .269 .028 -.181 -.296

.129 -.776 -.641 -.519 -.382 .641 .540 .269 .026 -.183

.141 .622 .516 .414 .304 .198 .129 .134 .138 .140

EV Split reg dbs

2

3

4

5

6

7

8

9

10

Ace

.043 .149

.018 .128

-.007 .107

-.031 .082

-.048 .074

-.006 .115

.150 .239

.319 .367

.510 .510

.634 .634

ApPENDIX

E

327

6 Decks, H17, Dealer Shows 5 EV Stand 2 Ace 10

9 8 7 6 5 4

3 2

-.157 -.161 -.161 -.160 -.159 -.155 -.154 -.155 -.155 -.156

3 -.157 -.161 -.160 -.159 -.159 -.155 -.154 -.154 -.155 -.155

4

-.156 -.160 -.160 -.159 -.158 -.154 -.154 -.154 -.154 -.155

5 -.156 -.160 -.159 -.159 -.158 -.154 -.153 -.154 -.154 -.154

6

7

8

9

-.039 -.161 -.160 -.159 -.159 -.155 -.154 -.154 -.155 -.155

.199 -.046 -.164 -.163 -.163 -.159 -.158 -.158 -.159 -.159

.440 .195 -.046 -.164 -.163 -.159 -.159 -.159 -.159 -.160

.670 .437 .195 -.046 -.164 -.160 -.159 -.160 -.160 -.161

10 .891 .669 .437 .195 -.046 -.161 -.160 -.160 -.161 -.161

-.159 .891 .670 .440 .199 -.039 -.156 -.156 -.157 -.157

Ace

EV Hit Ace

10 9 8 7 6 5 4

3 2

2

3

4

5

6

7

8

9

10

Ace

.138 -.189 .312 .262 .163 .079 .003 -.027 -.016 -.004

.116 -.254 -.194 .314 .264 .169 .081 .001 -.027 -.016

.095 -.321 -.254 -.192 .316 .269 .171 .083 .001 -.027

.075 -.389 -.321 -260 -.191 .320 .270 .171 .081 .003

.099 -.450 -.389 -.321 -.260 -.189 .320 .269 .169 .079

.151 -.519 -.451 -.384 -.330 -.260 -.191 .316 .264 .163

.206 -.612 -.519 -.451 -.384 -.321 -.260 -.192 .314 .262

.258 -.723 -.612 -.519 -.451 -.389 -.321 -.254 -.194 .312

.311 -.853 -.723 -.612 -.519 -.450 -.389 -.321 -.254 -.189

.161 .311 .258 .206 .151 .099 .075 .095 .116 .138

5 .130 -.777 -.642 -.520 -.382 .641 .540 .268 .026 -.181

6 7 8 9 10 .197 .302 .412 .515 .622 -.899 -1.038 -1.224 -1.447 -1.706 -.778 -.902 -1.038 -1.224 -1.447 -.643 -.768 -.902 -1.038 -1.224 -.519 -.659 -.768 -.902 -1.038 -.377 -.519 -.643 -.778 -.899 .641 -.382 -.520 -.642 -.777 .537 .632 -.385 -.508 -.642 .265 .528 .628 -.388 -.509 .022 .256 .524 .624 -.378

Ace

5 -.028 .084

6

7

-.005 .116

10 .509 .509

Ace

-.046 .076

EV Double Down Ace

10 9 8 7 6 5 4 3 2

2

3

4

.141 -.378 .624 .524 .256 .022 -.181 -.291 -.310 -.311

.139 -.509 -.388 .628 .528 .265 .026 -.180 -.291 -.310

.135 -.642 -.508 -.385 .632 .537 .268 .027 -.180 -.291

3 .020 .130

-.004 .109

.142 .622 .515 .412 .302 .197 .130 .135 .139 .141

EV Split 2 reg dbs

.045 .150

4

8 .148 .237

9

.318 .366

.634 .634

328



10 9 8 7

6 5 4

3 2

2

3

4

5

6

7

8

9

-.147 -.155 -.154 -.153 -.153 -.152 -.148 -.148 -.149 -.149

-.147 -.154 -.154 -.153 -.152 -.152 -.148 -.148 -.148 -.149

-.147 -.154 -.153 -.153 -.152 -.151 -.147 -.148 -.148 -.148

-.146 -.154 -.153 -.152 -.152 -.151 -.147 -.147 -.148 -.148

.012 -.158 -.157 -.156 -.156 -.155 -.151 -.151 -.152 -.152

.280 .008 -.158 -.157 -.156 -.156 -.152 -.152 -.152 -.153

.494 .281 .008 -.158 -.157 -.156 -.152 -.153 -.153 -.153

.703 .494 .281 .008 -.158 -.157 -.153 -.153 -.154 -.154

10 .902 .703 .494 .282 .008 -.158 -.154 -.154 -.154 -.155

Ace

-.146 .902 .703 .494 .280 .012 -.146 -,147 -.147 -.147

EV Hit Ace

10 9 8 7

6 5 4

3 2

2

3

4

S

6

7

8

9

.163 -.169 .336 .291 .199 .118 .036 -.009 .002 .014

.140 -.235 -.174 .337 .293 .202 .123 .034 -.009 .002

.118 -.302 -.234 -.173 .339 .295 .206 .124 .034 -.009

.101 -.369 -.302 -.241 -.172 .341 .299 .206 .123 .036

.129 -.425 -.371 -.304 -.243 -.174 .341 .295 .202 .118

.191 -.505 -.425 -.365 -.311 -.243 -.172 .339 .293 .199

.240 -.604 -.505 -.425 -.365 -.304 -.241 -.173 .337 .291

.287 -.720 -.604 -.505 -.425 -.371 -.302 -.234 -.174 .336

6

7

8

9

10 .335 -.852 -.720 -.604 -.505 -.425 -.369 -.302 -.235 -.169

Ace

10 .670 -1.705 -1.440 -1.208 -1.009 -.851 -.739 -.604 -.469 -.338

Ace

.188 .335 .287 .240 .191 .129 .101 .118 .140 .163

EV Double Down Ace

10 9 8 7 6 5 4

3 2

2 .188 -.338 .672 .582 .324 .093 -.126 -.272 -.297 -.298

3

4

5

.186 -.469 -.348 .675 .586 .327 .103 -.126 -.271 -.297

.183 -.604 -.469 -.345 .678 .590 .337 .103 -.126 -.272

.185 -.739 -.604 -.481 -.343 .683 .599 .337 .103 -.126

.258 .383 .480 .573 -.851 -1.009 -1.208 -1.440 -.743 -.851 -1.009 -1.208 -.608 -.730 -.851 -1.009 -.485 -.622 -.730 -.851 -.348 -.485 -.608 -.743 .683 -.343 -.481 -.604 .590 .678 -.345 -.469 .327 .586 .675 -.348 .093 .324 .582 .672

.191 .670 ,573 .480 ,383 ,258 .185 .183 .186 .188

EV Split reg dbs

2

3

4

S

6

7

8

9

10

Ace

.086 .208

.059 .184

.032 .161

.007 .136

-.022 .116

.059 .193

.230 .328

.387 .439

.567 .567

.682 .682

E

ApPENDIX

329

6 Decks, HI7, Dealer Shows 6 EV Stand 2 Ace

10 9 8 7 6 5 4 3 2

3

4

5

6

7

8

9

10

Ace

.894 .677 .451 .220 -.010 -.125 -.121 -.121 -.121 -.122

-.115 .894 .678 .452 .223 -.005 -.115 -.115 -.115 -.115

-.115 -.122 -.121 -.120 -.120 -.119 -.115 -.115 -.115 -.116

-.115 -.121 -.121 -.120 -.119 -.118 -.114 -.115 -.115 -.115

-.115 -.121 -.120 -.120 -.119 -.118 -.114 -.114 -.115 -.115

-.115 -.121 -.120 -.120 -.119 -.118 -.114 -.114 -.114 -.115

-.005 -.125 -.124 -.123 -.123 -.122 -.118 -.118 -.118 -.119

.223 -.010 -.125 -.124 -.124 -.123 -.119 -.119 -.119 -.120

.452 .220 -.010 -.125 -.124 -.123 -.120 -.120 -.120 -.120

.678 .451 .220 -.010 -.125 -.124 -.120 -.120 -.121 -.121

2

3 .146 -.239 -.176 .336 .287 .193 .111 .036 .010 .019

4

5 .111 -.379 -.309 -.245 -.173 .340 .294 .198 .111 .037

6 .126 -.438 -.381 -.311 -.247 -.175 .340 .289 .193 .107

7

10

Ace

.178 -.515 -.438 -.375 -.318 -.247 -.173 .338 .287 .190

8 .231 -.610 -.515 -.438 -.375 -.311 -.245 -.174 .336 .286

9

.126 -.309 -.239 -.174 .338 .289 .198 .113 .036 .010

.281 -.723 -.610 -.515 -.438 -.381 -.309 -.239 -.176 .334

.333 -.853 -.723 -.610 -.515 -.438 -.379 -.309 -.239 -.171

.190 .333 .281 .231 .178 .126 .111 .126 .146 .167

7 6 10 9 8 .252 .357 .462 .563 .667 -.875 -1.030 -1.220 -1.445 -1.706 -.762 -.875 -1.031 -1.220 -1.445 -.622 -.750 -.875 -1.031 -1.220 -.493 -.636 -.750 -.875 -1.030 -.350 -.493 -.622 -.762 -.875 .680 -.346 -.490 -.618 -.758 .579 .676 -.349 -.477 -.617 .315 .575 .672 -.352 -.478 .082 .311 .571 .669 -.341

Ace

10 .556 .556

Ace

EV Hit Ace

10 9 8 7 6 S 4 3 2

.167 -.171 .334 .286 .190 .107 .037 .010 .019 .030

EV Double Down Ace

10 9 8 7 6 5 4 3 2

2

3

4

.204 -.341 .669 .571 .311 .082 -.105 -.212 -.231 -.231

.203 -.478 -.352 .672 .575 .315 .092 -.105 -.212 -.231

.200 -.617

-.4n -.349 .676 .579 .324 .093 -.105 -.212

5

.202 -.758 -.618 -.490 -.346 .680 .587 .324 .092 -.105

.206 .667 .563 .462 .357 .252 .202 .200 .203 .204

EV Split 2

reg dbs

.111 .234

3 .087 .213

4

.064 .193

5 .041 .169

6 .013 .150

7

.061 .198

8 .207 .303

9 .370 .421

.680 .680

330


6 Decks, Dealer Shows 7 EV Stand Ace

10 9 8 7 6

5 4 3 2

2

3

4

6 -.104 -.477 -.480 -.479 -.478 -.478 -.477 -.476 -.476 -.476

8

9

10

Ace

-.473 -.475 -.478

5 -.474 -.476 -.479 -.478 -.478 -.477 -.476 -.475 -.475 -.475

7

-.473 -.475 -.478 -.477

.402 -.109 -.481 -.480 -.479 -.478 -.478 -.476 -.477 -.477

.616 .398 -.110 -.480 -.480 -.479 -.478 -.477 -.477 -.477

.773 .615 .400 -.110 -.481 -.480 -.479 -.478 -.478 -.478

.926 .772 .615 .398 -.109 -.477 -.476 -.474 -.475 -.475

-.472 .926 .773 .616 .402 -.104 -.474 -.473 -.473 -.473

-.476 -.475 -.474 -.474 -.475

-.477 -.476 -.475 -.474 -.474 -.474

-.473 -.474 -.478 -.477 -.476 -.476 -.475 -.473 -.474 -.474

2

3

4

5

6

7

8

9

10

Ace

.120 -.213 .292 .259 .174 .084 -.068 -.153 -.119 -.089

.077 -.269 -.218 .292 .260 .176 .084 -.069 -.154 -.119

.036 -.325 -.270 -.218 .293 .262 .176 .087 -.069 -.153

-.008 -.369 -.325 -.278 -.220 .293 .260 .176 .084 -.068

.055 -.409 -.369 -.325 -.278 -.221 .293 .262 .176 .084

.172 -.478 -.408 -.363 -.332 -.278 -.220 .293 .260 .174

.221 -.587 -.478 -.408 -.363 -.325 -.278 -.218 .292 .259

.255 -.713 -.587 -.478 -.408 -.369 -.325 -.270 -.218 .292

.291 -.850 -.713 -.587 -.478 -.409 -.369 -.325 -.269 -.213

.164 .291 .255 .221 .172 .055 -.008 .036 .077 .120

7

8

9

10

Ace

.224 .321 .386 .464 -.957 -1.175 -1.426 -1.701 -.817 -.956 -1.174 -1.426 -.736 -.817 -.956 -1.175 -.692 -.736 -.817 -.957 -.607 -.678 -.748 -.817 -.518 -.606 -.677 -.747 .468 -.516 -.589 -.674 .401 .464 -.518 -.587 .114 .396 .462 -.505

-.176

-.4n

-.4n

EV Hit Ace 10 9 8 7 6

5 4 3 2

EV Double Down Ace

10 9 8 7 6 S 4

3 2

2

3

4

5

6

-.180 -.505 .462 .396 .114 -.181 -.586 -.890 -.949 -.949

-.182 -.587 -.518 .464 .401 .116 -.179 -.582 -.890 -.949

-.177 -.674 -.589 -.516 .468 .406 .118 -.175 -.582 -.890

-.184 -.747 -.677 -.606 -.518 .470 .405 .118 -.179 -.586

-.009 -.817 -.748 -.678 -.607 -.520 .470 .406 .116 -.181

3 -.115 -.066

4

5

6

7

8

9

10

-.183 -.133

-.248 -.199

-.299 -.249

-.135 -.086

.171 .218

.343 .369

.509 .509

.464 .386 .321 .224

-.009 -.184 -.177 -.182 -.180

EV Split 2

reg dbs

-.054 -.005

Ace .476 .476

ApPENDIX

E

331

6 Decks, Dealer Shows 8 EV Stand Ace 10 9 8 7

6 5 4 3 2

2

3

4

5

6

7

-.510 -.512 -.511 -.514 -.514 -.513 -.512 -.512 -.511 -.511

-.510 -.511 -.511 -.514 -.513 -.513 -.512 -.512 -.511 -.511

-.511 -.513 -.512 -.516 -.515 -.514 -.513 -.513 -.512 -.512

-.511 -.513 -.512 -.515 -.515 -.514 -.513 -.513 -.512 -.512

-.383 -.513 -.513 -.516 -.515 -.515 -.514 -.514 -.513 -.513

.108 -.384 -.513 -.517 -.516 -.515 -.515 -.515 -.513 -.514

2

3

4

5

6

7

.052 -.272 .228 .199 .100 -.059 -.212 -.220 -.187 -.157

.017 -.234 -.279 .229 .201 .101 -.059 -.213 -.219 -.187

-.028 -.370 -.326 -.279 .229 .200 .100 -.059 -.213 -.220

-,069 -.417 -.372 -.333 -.279 .230 .200 .100 -.059 -.212

-.072 -.453 -.417 -.371 -.335 -.279 .230 .200 .101 -.059

.041 -.501 -.454 -.411 -.378 -.335 -.279 .229 .201 .100

9 .791 .591 .099 -.388 -.513 -.513 -.512 -.512 -.511 -.511

10 .930 .790 .591 .104 -.384 -.513 -.513 -.513 -.511 -.512

Ace -.509 .930 .791 .596 .108 -.383 -.511 -.511 -.510 -.510

8

9

.194 -.711 -.587 -.501 -.454 -.417 -.372 -.326 -.279 .228

10 .229 -.850 -.711 -.587 -.501 -.453 -.417 -.370 -.324 -.272

Ace

.153 -.587 -.501 -.453 -.411 -.371 -.333 -.279 .229 .199

8 .596 .104 -.388 -.518 -.517 -.516 -.515 -.516 -.514 -.514

EV Hit Ace 10 9 8 7

6 5 4 3 2

.095 .229 .194 .153 .041 -.072 -.069 -.028 .017 .052

EV Double Down 2

Ace 10 9

8 7

6 5 4 3 2

3

4

5

-.314 -.305 -.314 -.617 -.693 -.762 .347 -.630 -.696 .288 .347 -.632 -.022 .293 .349 -.450 -.020 .292 -.848 -.448 -.022 -1.005 -.847 -.451 -1.022 -1.001 -.847 -1.023 -1.022 -1.005

-.317 -.842 -.765 -.713 -.631 .353 .293 -.022 -.448 -.848

7 6 8 9 10 -.251 -.028 .194 .277 .348 -.906 -1.002 -1.174 -1.422 -1.700 -.844 -.907 -1.002 -1.174 -1.422 -.768 -.831 -.907 -1.002 -1.174 -.715 -.781 -.831 -.907 -1.002 -.631 -.715 -.768 -.844 -.906 .353 -.631 -.713 -.765 -.842 .292 .349 -.632 -.696 -.762 -.020 .293 .347 -.630 -.693 -.450 -.022 .288 .347 -.617

Ace

-.312 .348

.2n

.194 -.028 -.251 -.317 -.314 -.305 -.314

EV Split reg dbs

2

3

-.207 -.174

-.263 -.230

4 -.322 -.288

5

6

7

8

9

10

Ace

-.387 -.354

-.432 -.398

-.422 -.389

-.116 -.085

.193 .212

.389 .389

.360 .360

332


6 Decks, Dealer Shows 9 EV Stand 2 -.539 -.541 -.540 -.540 -.543 -.542 -.541 -.542 -.542 -.540

Ace

10 9 8 7 6 5 4 3 2

3 -.541 -.542 -.542 -.541 -.544 -.543 -.543 -.543 -.543 -.542

4 -.540 -.542 -.541 -.541 -.544 -.543 -.543 -.543 -.543 -.542

5 -.540 -.542 -.541 -.541 -.544 -.543 -.543 -.543 -.543 -.541

6 -.421 -.543 -.542 -.541 -.545 -.544 -.543 -.543 -.543 -.542

7

-.183 -.422 -.543 -.542 -.546 -.545 -.544 -.544 -.544 -.543

8 9 .288 .759 -.185 .284 -.421 -.185 -.539 . -.421 -.542 -.543 -.541 -.542 -.541 -.541 -.541 -.541 -.541 -.542 -.540 -.540

10 .939 .756 .284 -.185 -.422 -.543 -.542 -.542 -.542 -.541

Ace

-.538 .939 .759 .288 -.183 -.421 -,540 -,540 -.541 -.539

EV Hit Ace

10 9

8 7 6 5 4

3 2

2

3

4

5

6

7

8

9

-.034 -.341 .156 .117 -.052 -.210 -.285 -.294 -.266 -.238

-.073 -.383 -.348 .155 .117 -.052 -.211 -.288 -.295 -.266

-.112 -.428 -.384 -.349 .157 .117 -.052 -.210 -.288 -.294

-.151 -.472 -.430 -.391 -.348 .157 .117 -.052 -.211 -.285

-.147 -.505 -.473 -.432 -.392 -.347 .157 .117 -.052 -.210

-.098 -.549 -.505 -.467 -.438 -.392 -.348 .157 .117 -.052

.007 -.613 -.550 -.506 -.467 -.432 -.391 -.349 .155 .117

.113 -.713 -.613 -.550 -.505 -.473 -.430 -.384 -.348 .156

10 .157 -.849 -.713 -.613 -.549 -.505 -.472 -.428 -.383 -.341

.000 .157 .113 .007 -.098 -.147 -.151 -.112 -.073 -.034

7 5 6 8 9 10 -.456 -.392 -.285 -.071 .139 .226 -.949 -1.009 -1.099 -1.226 -1.426 -1.699 -.875 -.951 -1.010 -1.100 -1.226 -1.426 -.814 -.879 -.940 -1.011 -1.100 -1.226 -.752 -.818 -.894 -.940 -1.010 -1.099 .230 -.750 -.818 -.879 -.951 -1.009 .149 .230 -.752 -.814 -.875 -.949 -.297 .148 .227 -.753 -.800 -.872 -.716 -.299 .146 .226 -.754 -.799 -.952 -.714 -.297 .149 .225 -.739

-.451 .226 .139 -.071 -.285 -.392 -.456 -.451 -.446 -.443

Ace


10 9 8 7 6 5

4

3 2

3

4

-.443 -.446 -.451 -.739 -.799 -.872 .225 -.754 -.800 .149 .226 -.753 -.297 .146 .227 -.714 -.299 .148 -.952 -.716 -.297 -1.064 -.955 -.716 -1.083 -1.067 -.955 -1.080 -1.083 -1.064

Ace

EV Split reg dbs

2

3

4

S

6

7

-.380 -.364

-.429 -.413

-.483 -.467

-.544 -.528

-.583 -.567

-.570 -.555

8 -.421 -.405

9

10

Ace

-.106 -.095

.225 .225

.238 .238

ApPENDIX

E

333

6 Decks, Dealer Shows 10 EV Stand 2

3 -.540 -.540 -.540 -.539 -.539 -.542 -.541 -.541 -.541 -.542

4

5 -.540 -.540 -.540 -.539 -.538 -.542 -.541 -,541 -.541 -.541

6

7

-.420 -.541 -.540 -.540 -.539 -.543 -.542 -.542 -.542 -.542

-.180 -.418 -.537 -.536 -.536 -.539 -.538 -.538 -.539 -.539

5 -.211 -.504 -.464 -.429 -.382 .118 .027 -.151 -.250 -.320

6

7

-.196 -.535 -.505 -.466 -.429 -.382 .118 .027 -.151 -.250

-.143 -.580 -.535 -.500 -.474 -.429 -.382 .117 .026 -.153

10 .962 .559 .069 -.175 -.418 -.541 -.540 -.540 -.540 -.540

Ace -.539 .962 .555 .063 -.180 -.420 -.540 -.540 -.540 -.540

9

10

Ace

.023 -.726 -.644 -.580 -.535 -.505 -.464 -.421 -.382 .117

.117 -.847 -.726 -.644 -.580 -.535 -.504 -.463 -.420 -.376

-.066 .117 .023 -.088 -.143 -.196 -.211 -.173 -.137 -.102

5 6 7 8 9 10 -.509 -.509 -.511 -.518 -.455 -.343 -.234 -.015 .173 -.788 -.859 -.932 -1.008 -1.069 -1.160 -1.287 -1.453 -1.694 .174 -.801 -.860 -.934 -1.011 -1.071 -1.161 -1.288 -1.453 -.006 .177 -.798 -.874 -.938 -1.000 -1.071 -1.161 -1.287 -.462 -.005 .178 -.796 -.874 -.954 -1.000 -1.071 -1.160 -.747 -.464 -.006 .178 -.798 -.874 -.938 -1.011 -1.069 -.951 -.745 -.462 -.004 .178 -.796 -.874 -.934 -1.008 -1.064 -.951 -.745 -.462 -.006 .178 -.798 -.860 -.932 -1.083 -1.064 -.951 -.745 -.464 -.005 .177 -.801 -.859 -1.083 -1.083 -1.064 -.951 -.747 -.462 -.006 .174 -.788

Ace -.507 .173 -.015 -.234 -.343 -.455 -.518 -.511 -.509 -.509

Ace 10

-.540 -.540 -.540 -.539 -.539 -.542 -.541 -.541 -.542 -.542

9 8 7 6 5 4

3 2

-.540 -.540 -.540 -.539 -.538 -.542 -.541 -.541 -.541 -.541

8 .063 -.175 -.415 -.537 -.536 -.540 -.539 -.539 -.539 -.539

.555 .069 -.171 -.415 -.537 -.540 -.540 -.540 -.540 -.540

8 -.088 -.644 -.580 -.535 -.500 -.466 -.429 -.382 .117 .026

9

EV Hit Ace 10 9 8 7 6 5 4

3 2

2

3

4

-.102 -.376 .117 .026 -.153 -.250 -.32G -.339 -.313 -.287

-.137 -.420 -.382 .117 .026 -.151 -.250 -.322 -.339 -.313

-.173 -.463 -.421 -.382 .117 .027 -.151 -.249 -.322 -.339

EV Double Down Ace 10 9 8 7 6 5 4

3 2

2

3

4

2

3 -.527 -.518

-.578 -.568

EV Split reg dbs

-.480 -.471

4

5 -.635 -.625

6

7

-.672 -.662

-.638 -.628

8 -.493 -.483

9

10

-.299 -.290

.052 .052

Ace .182 .182

334


GLOSSARY

Blackjack has its own vocabulary. This glossary defines all of the blackjack terminology used in this book. Some commonly used words have two meanings in blackjack conversations, and this glossary explains which of the meanings is used in this book. Ace. An ace is a card that you may count as either one or eleven. The easiest way to count a hand containing an ace is to count the ace as eleven initially. If your hand total comes to more than twenty-one with the ace as eleven, then count the ace as one. Ace adjustment. If you count aces in addition to your primary count, when you use your count of aces you are said to be making an ace adjustment. Action. Your action is the total amount you bet on all of your hands. If you play 100 hands at $25 each, then your action is $2500. Back count. To back count is to count cards at a table as a spectator, Le., without playing a hand. Balanced count. This is a counting system in which the positive cards exactly balance the negative cards, so the total count over a complete deck is zero.

GLOSSARY

335

Basic strategy. Basic strategy for blackjack is the set of playing decisions you should make in order to maximize your advantage if you use no information about cards remaining in the pack except that the cards in your hand and the upcard are not among them. You are presumed to know the number of decks shuffled together, the details of the rules, and what procedures the dealer must follow. Bet variation. In order to get an edge at blackjack, you generally have to bet more when you have an edge and less when the casino has an edge. The ratio of your big bet to your small bet is the bet variation. Black check. Most casinos use black as the color of their $100 chips. Black action means bets with $100 chips. Blackjack. Blackjack is the name of the game. In conversation but not in this book, blackjack also means a natural. Bonus. Some casinos pay bonuses for certain hands, such as hands containing six cards or 6-7-8 all in the same suit. Break the deck. To break the deck is to shuffle earlier than usual. This sometimes is done if a customer is suspected to be a card counter. It is also called shuffle up. Burn. After the shuffle and cut, one card is placed on the bottom of the pack or in the discard tray. This procedure is called burning a card, and the card temporarily removed from play is called the burned card or bum card. Bust. To bust is to go over twenty-one. If you bust you lose. If the dealer busts and you do not bust, you win. If both you and the dealer bust, you lose; this is the basis for the casino's edge. Bustout. bet. This is a side bet offered at some casinos. You can make it after the dealer's second card is shown, when the dealer has a total of twelve through sixteen. When you make the bustout bet, you are betting that the next card will be a IO-count card. If it is, your bustout bet wins 2: I. Camouflage. To use camouflage is to make a bad play in hopes of looking a gambler rather than a card counter. Camouflage costs money, so if you do too much of it you give up your edge. Check. A check or a chip is a form of money in a casino. Chip. See check. Counting cards. Card counting means keeping some sort of track of which cards have been played. Card counters gener-

336


ally vary bet size and playing strategy as the composition of the remaining pack changes. Count per deck. Count per deck is the running count divided by the number of decks or fraction of a deck that you have not seen. Never use this expression in a casino. Crush count. 9s and lOs count -1; aces and 2s count +2; 3s count +1; and 4s through 8s count zero. This count is used for taking advantage of the over/under 13 bets. Cut. When you cut the pack you divide it into two parts. The dealer then inverts the order of the two parts. Deal. To deal is to distribute the cards. Dealer. The dealer is the casino employee who deals the cards. Deck. This book uses deck to mean 52 cards. A single deck contains four aces, four each of 2 through 9, and sixteen lOs. In conversation but not in this book, deck is also used to mean pack. Discards. Discards are the cards used since the last shuffle. They are set aside to be brought into play at the next shuffle. Double down. When you double down, if your cards are not already face up you must turn them up. Then double your bet. You will receive one more card. The advantage of doubling down is that you are allowed to increase your bet after you see your hand and the dealer's upcard. The disadvantage is that you must take one and only one more card. Doubling down after splitting. Suppose you split 8-8 against 5. If one of your hands catches 3 for eleven, you should like to double down. At some casinos you are allowed to double down after splitting a pair, and at others you are not. Double exposure. The dealer gets two cards face up before you play your hand. Early surrender. Early surrender is when you may surrender before the dealer checks for a natural. If the dealer turns over a natural after you have early-surrendered, the early surrender saved you half a bet. Einstein count. See hi-opt. Even money. When you have a natural, accepting even money is the same as putting out an insurance bet. So even money is another term for insurance, but only if you have a natural.

GLOSSARY

337

Expected win. The average result if you were able to make a bet over and over again in the same apparent situation is the expected win of that bet. First base. You are at first base if you are playing the spot nearest the dealer's left hand. The dealer finishes hitting your hand before moving on to the hands of the other players. Five-card half win. If you have five cards and you have not busted, you have the option of turning in your hand and accepting a win equal to half of your bet. This rule is offered at some Asian casinos. Half win. See five-card half win. Halves count. Aces and lOs count -1; 9s count -0.5; 2s and 7s count +0.5; 3s, 4s, and 6s count +1; 5s count +1.5; and 8s count zero. Hand. The dealer gives you two cards to start your hand, and you may take more cards if you wish. All these cards are your hand. The dealer's cards are the dealer's hand. In conversation but not in this book, hand is also used to mean round. Hard. Any hand that does not contain an ace is a hard hand. A hand with an ace can be hard too, if the hand totals twelve or more and all aces in the hand count one. Examples: 6-7 is hard thirteen; 5-10-ace is hard sixteen. Since twelve is the smallest soft hand, the word "hard" is not used with hands of eleven or less. High-low count. Aces and lOs count -1, 2s through 6s count +1, and 7s through 9s count zero. Hi-opt or hi-opt I. lOs count -1; 3s through 6s count +1; and aces, 2s, 7s, 8s, and 9s count zero. Also known as the Einstein count. Sometimes hi-opt I is used to refer to two counts, this one and a side count of aces. In this book, hi-opt is used to mean the primary count only, with no side count of aces. There also is a hi-opt II, a higher-level count. Hit. To hit a hand is to take a card. The card you receive when you hit is also called a hit. Hole card. When the dealer gets a card face up and a card face down, the hole card is the card that is face down. You are not supposed to know the value of the hole card until after you play your hand. Index number. See strategy number.

338


Intermediate. An intermediate card is a 7, 8, or 9. lOs and aces are called big cards and 2s, 3s, 4s, 5s, and 6s are called small cards. Insurance. Insurance is a side bet that the dealer has a natural. You may buy insurance only when the dealer's upcard is an ace. If the dealer has a natural, the insurance bet wins double. If the dealer does not have a natural, the insurance bet loses. Kelly betting. This means betting a percentage of your bankroll equal to your advantage divided by· the variance of possible outcomes. It is also known as proportional betting. Late surrender. Late surrender is when you must wait to surrender until after the dealer checks the hole card to see if the hand is a natural. If the dealer has a natural you may not surrender. Lay and pay. This is one procedure a dealer might use to pick up the cards. It involves leaving all the cards on the table until the payouts have been completed. An alternate procedure is pick and pay. Lose all to a natural. When you double down or split and the dealer winds up with a natural, you lose both bets. This rule is common in Europe. Lose one bet to a natural. When you double down or split and the dealer winds up with a natural, you lose only your original bet. Macao surrender. See five-card half win. Multiple deck. A multiple-deck game is blackjack using more than one deck. Natural. A natural is a two-card hand of twenty-one points. The only way to make twenty-one in two cards is ace-IO. In conversation but not in this book, a natural is sometimes called a blackjack. Over/under 13. This is two side bets. Aces count one and thirteen loses. On the over you bet that your first two cards will total over thirteen. On the under you bet that they will total under thirteen Pack. This book uses pack to denote any collection of cards, as distinguished from deck, which is 52 cards. Pair. A pair is two cards of the same value, such as 7-7 or 8-8. Whether two unlike lOs (such as queen-king) are a pair is up

GLOSSARY

339

to the casino management. Generally a pair may be split and played as two hands.

Pat. A pat hand is an unbusted hand worth at least seventeen points. A dealer or player who has a pat hand is said to be pat. Penetration. This word describes how far down into the pack the dealer goes before shuffling. The better the penetration, the more profit potential for a card counter, everything else being equal. Pick and pay. This is one procedure a dealer might use to pick up the cards. It involves picking up the cards on a hand as soon as the bet on that hand has been settled. An alternate procedure is lay and pay. Pit. Blackjack tables are not randomly scattered throughout the casino. Rather, they are arranged in groups called pits. Pit boss. A pit boss is the casino employee in charge of all the tables in a particular pit. Proportional betting. Proportional betting is betting a percentage of your bankroll equal to your advantage divided by the variance of possible outcomes. It is also known as Kelly betting. Push. A push is a tie, and no money changes hands. A push occurs when both you and the dealer have unbusted hands with the same total points. If you bust you cannot push. If ties lose, as they do in double exposure, there are no pushes. Random. For the cards in a deck to be random, each card must have an equal chance of being in each position, and all possible orderings of the cards must be equally likely. Rime. A rifile is part of a shuffle. A ritlle is taking two like-sized groups of cards and interleaving one into the other. Resplit. The opportunity to resplit occurs when you split a pair and receive another card of the same value. To resplit means to put out a third bet and play the cards as three hands. You sometimes have opportunities for further resplits to make four or more hands. Round. One round includes making bets, dealing two cards to each player and the dealer, serving the players' hands, finishing the dealer's hand, and making the payoffs.

340


Running count. The running count is the total points your counting system assigns to all the cards you have seen since the last shuffle. Never use this expression in a casino. Second-chance 21. This rarely-offered rule allows you to throw in your tirst hand and receive a new hand, at the cost of half a bet. Shoe. A shoe is a box for holding cards. Since dealing with more than two decks held in the hand is awkward, three or more decks are dealt out of a shoe. Sometimes one or two decks are dealt from a shoe. Shoe game. A shoe game is a blackjack game in which the dealer deals out of a shoe. Shuffle. This book uses shuffle to mean the sum total of all actions the dealer uses to reorder the cards. Outside of this book, a shuffle is sometimes used to mean a riffle. Shuffle up. See break the deck. Single exposure. This is the traditional variety of blackjack the dealer's hand has one card face up for you to see before you play your hand. Snapper. Another word for natural. Soft. A soft hand contains at least one ace and is valued at twentyone or less when one ace is counted as eleven. Examples: Ace6 is soft seventeen, and ace-6-ace is soft eighteen. The smallest soft hand possible is twelve, and the largest is twenty-one. Split. To split you must have a pair. If your cards are not already face up, tum them up. Then put out an additional bet equal to your original bet, and play the cards as two hands. Stand. To stand is to take no more cards. Stiff. A stiff is a hand that is not pat and that may bust if hit once. Stiffs include hard twelve through sixteen. In conversation but not in this book, stiff has another meaning: a person who does not tip. Strategy number. Comparison of your count per deck to a strategy number tells you whether to hit or stand, or double down, etc. Strip. A strip is part of a shuffle. The dealer holds the deck with one hand and pulls groups of cards off the top with the other

GLOSSARY

341

hand, with the result that the order of the groups in inverted. A strip can also mean a street containing multiple casinos, and when capitalized means Las Vegas Boulevard. Surrender. To surrender is to give up half of your bet for the privilege of not playing out your hand. You keep the other half of your bet. TARGET. This is an acronym that stands for nonsense. Proponents of TARGET think you can extrapolate past results such as wins and losses and whether the dealer has been busting to predict future results. Team play. This means two or more people playing blackjack in the same casino and sharing information. It might mean two card counters at the same table but using different systems to help each other make better decisions. Or it might be a card counter betting small sums who, when the count is high, calls in a partner who makes only big bets. Third base. You are at third base if you have the spot to the dealer's right. Yours is the last player hand to be finished. Tie. A tie is when you and the dealer end up with the same total points. True count. This is another way of saying count per deck. Twenty-one. This is another name for the game of blackjack. Unbalanced count. This is a counting system in which the total points in a full deck do not add up to zero. Upcard. The upcard is the card in the dealer's hand that is face up for all the players to see before they play their hands. Win rate. Win rate is how fast you are expected to win. It can be expressed in percent or in dollars per hour.

342


SELECTED -REFERENCES

Andersen, Ian. Turning the Tables on Las Vegas. New York: The Vanguard Press, 1976. Baldwin, Roger, Wilbert Cantey, Herbert Maisel, and James McDermott. "The Optimum Strategy in Blackjack," Journal of the American Statistical Association. 51 (1956); 429-439. Baldwin, Roger, Wilbert Cantey, Herbert Maisel, and James McDermott. Playing Blackjack to Win. New York: M. Barrows & Co., 1957. Bourie, Steve. American Casino Guide. Dania, FL: Casino Vacations, 1997. Braun, Julian H. How to Play Winning Blackjack. Chicago: Data House Publishing Co. Inc., 1980. Canfield, Richard Albert. Blackjack Your Way to Riches. Scottsdale: Expertise Publishing Co., 1977. Card Player, The. Las Vegas: Card Player Limited, biweekly since 1988.

SELECTED REFERENCES

343

Carlson, Bryce. Blackjack for Blood. Santa Monica: CompuStar Press, 1992; last revised 1997. Casino Player. Atlantic City: ACE Marketing, Inc., monthly since 1989. Chambliss, C. R. and T. C. Roginski. Fundamentals of Blackjack. Las Vegas: GBC, 1990. Chambliss, C. R. and T. C. Roginski. Playing Blackjack in Atlantic City. Las Vegas: GBC, 1981. Curtis, Anthony. Bargain City. Las Vegas: Huntington Press, 1993. Curtis, Anthony. Las Vegas Advisor. Las Vegas: Huntington Press, monthly since 1983. Dalton, Michael. Blackjack: A Professional Reference. Merritt Island, FL: Spur of the Moment Publishing, 1991; last revised 1993. Dalton, Michael. Blackjack Review. Merritt Island, FL: Spur of the Moment Publishing, quarterly since 1992. Epstein, Richard A. The Theory of Gambling and Statistical Logic. New York: Academic Press, 1977. Forte, Steve. Gambling Protection Series (videotapes). Las Vegas: BTA Joint Venture Inc., 1984-1987. Forte, Steve. Read the Dealer. Berkeley: RGE, 1986. Friedman, Bill. Casino Management. Secaucus, NJ: Lyle Stuart, 1982. Gates, R. W. Blackjack Monthly. Escondido, CA: Richard Canfield & Associates, monthly 1987-1995. ~, Griffin, Peter. Extra Stuff. Las Vegas: Huntington Press, 1991.

Griffin, Peter. The Theory of Blackjack. Las Vegas: Huntington Press, 1979; last revised 1996. Gwynn, John M. Jr. and Armand Seri. "Experimental Comparisons of Blackjack Betting Systems," presented at the Fourth Gambling Conference, 1978. Humble, Lance and Carl Cooper. The World s Greatest Blackjack Book. New York: Doubleday, 1980.

344


Livingston, A. D. Dealing With Cheats. Philadelphia: J. B. Lippincott Company, 1973. Malmuth, Mason. Blackjack Essays. Las Vegas: Mason Malmuth, 1987; last revised 1991. Perry, Stuart. Las Vegas Blackjack Diary. Pittsburgh: Conjelco, 1997. Revere, Lawrence. Playing Blackjack as a Business. Secaucus, NJ: Lyle Stuart, 1971. Rose, Nelson. Gambling and the Law. Hollywood: Gambling Times Inc., 1986. Rubin, Max. Comp City. Las Vegas: Huntington Press, 1994. Schlesinger, Don. Blackjack Attack. Oakland: RGE, 1997. Snyder, Arnold. Blackbelt in Blackjack. Oakland: RGE, 1983. Snyder, Arnold. Blackjack Forum. Berkeley: ROE, quarterly since 1981. Snyder, Arnold. The Blackjack Formula. Oakland: ROE, 1980. Snyder, Arnold. The Over/Under Report. Oakland: ROE, 1989. Thorp, Edward O. Beat the Dealer. New York: Random House, 1962; revised 1966. Uston, Ken. Million Dollar Blackjack. Hollywood: SRS Enterprises, 1981. Uston, Ken. The Big Player. New York: Holt, Rinehart and Winston, ] 977. Uston, Ken. Two Books on Blackjack. Wheaton, MD: Uston Institute of Blackjack, 1979. Vancura, Olaf & Ken Fuchs. Knock-Out Blackjack. Unstated: Isochoric Publishing, 1996. Wilson, Allan N. The Casino Gambler s Guide. New York: Harper & Row, 1966. Wong, Stanford. Basic Blackjack. La Jolla: Pi Vee Press, 1992; last revised 1993. Wong, Stanford. Betting Cheap Claimers. La Jolla: Pi Vee Press, 1992.

SELECTED REFERENCES

345

Wong, Stanford. Blackjack Analyzer (software). La Jolla: Pi Vee Press, 1992. Wong, Stanford. Blackjack Count Analyzer (software). La Jolla: Pi Vee Press, 1993. Wong, Stanford. Blackjack in Asia. La Jolla: Pi Vee Press, 1979. Wong, Stanford. Blackjack Secrets. La Jolla: Pi Vee Press, 1993. Wong, Stanford. Blackjack World. La Jolla: Pi Vee Press, monthly 1979-1984. Wong, Stanford. Casino Tournament Strategy. La Jolla: Pi Vee Press, 1992; last revised 1966. Wong, Stanford. Current Blackjack News. La Jolla: Pi Vee Press, monthly since 1979. Wong, Stanford. Nevada Blackjack. La Jolla: Pi Vee Press, monthly 1979-1984. Wong, Stanford. Optimal Strategy for Pai Gow Poker. La Jolla: Pi Vee Press, 1990; last revised 1992. Wong, Stanford. Professional Video Poker. La Jolla: Pi Vee Press, 1988; last revised 1993. Wong, Stanford. Winning Gamer. La Jolla: Pi Vee Press, monthly 1984-1986. Wong, Stanford. Winning Without Counting. La Jolla: Pi Vee Press, 1978. Zender, Bill. Card Counting for the Casino Executive. Las Vegas: Zender, 1990.

346


INDEX Symbols 10-count

54-57

A all-time high 200 automatic win 119-124

B Basic Blackjack 23, 24 Basic Blackjack Betting 239 basic strategy 22-29, 158159 basic strategy win rate 50 Beat the Dealer 31, 149, 219-220, 298 benchmark defined 18 benchmark win rate 19, 44 Blackjack Count Analyzer 4, 34, 36-37, 99, 166, 286287, 292 Blackjack for Blood 213 Blackjack Forum 50, 195, 248 Blackjack Monthly 145 Blackjack Secrets 4, 30, 213, 214

Blackjack World 5 bonus for 6-7-8 126-138 bonus for 7-7-7 138 Braun, Julian 31, 253, 293 bustout bet 56-57, 185

C Carlson, Bryce 213 Chambliss, Carlson R. 293 composition-dependent 2324 computer 212 Cooper, Carl 213 count per deck 32-34 crush count 145-147 Current Blackjack News

34 Curtis, Anthony

5,

5

D Dalton, Michael 5 dealer hits soft 17 44, 48, 182, 185 double down 25, 40, 58 double exposure 156-182

347

INDEX Douglas, Dave 5, 224-231 Dubner, Harvey 31

K Kelly criterion

203-205

E

L

early half-win 105 early surrender 90 Einstein, Charles 239 European rules 81-85 expected value 187-188 expected win 183-189

late surrender 92 lose all to a natural 185

F five-card half win 99-105, 185 French, Marvin 200

G Gaem, Bill 223 generic basic strategy 24-29 Giles, Peter 199, 224 Griffin, Peter 23, 3 1, 34, 53, 150, 200, 214, 250, 301 Gwynn, John M. Jr. 201, 246

H halves count 148-151, 174175, 185, 215, 270 hard hand 28, 43 hi-opt 174, 185, 213-215 high count 216 How to Play Winning Blackjack 31, 293 Humble, Lance 213 incomplete shuffle 223 insurance 29, 37, 52-57, 185

M multiple-card surrender 95

36

94-

N nadir 195-199 Nevada Blackjack 5 no dealer natural 85-88 no hole card 80-88, 90, 92 normal curve 192-200

o Olsen, Eddie 239 optimal bet size 201-206 optimal number of simultaneous hands 208-211 over/under 13 16, 142-147, 185 "Over/Under Report, The," 145 overbetting 206-207

p Patterson, Jerry 239 player edge 35, 157 Playing Blackjack as a Business 213, 293 Playing Blackjack in Atlantic City 293 Polo, Frank 5 proportional betting 20 I

R

J jumping into a game

81-85,

resolving toss-ups 218 resplits 68, 182, 185

348


Revere, Lawrence 31, 213, 293 Review Journal 199 risk 186-187, 189-200 Roginski, Thomas C. 293 running count 32

S sampling error 19-20 Schlesinger, Donald 5, 50, 199-200 second-chance 21 96-97 Seri, Armand 201 seven cards 98-124 shuffle 222-251 shuffle tracking 212 side counts 217 six cards 98-124, 185 Smallwood, Jake 145 Snyder, Arnold 53, 145146, 246 soft hand 25-28, 42 Speer, John 5 split 24-25, 38-39, 67-69 standard casino shuffle 224231 standard deviation 190-192 Stanford Wong's Blackjack JVelVsletter 5 stopping rules 207-208 streakiness 239-251 surrender 28, 89-97, 185 surrender after doubling down 96

T TARGET 239 Theory of Blackjack, The 23, 31, 34, 150, 301 Thorp, Edward O. 31, 149150, 219-220, 298-300 total-dependent 23-24 two-card combinations 217218

u Uston, Ken

199

W win rates 17-21, 185 win rates, double exposure 182 World's Greatest Blackjack Book, The 2 I3

349

PUBLICATIONS BY STANFORD WONG Books Professional Blackjack. This 350-page book has a complete and accurate presentation of the high-low, the counting system used by more card counters than any other because of its combination of simplicity and power. If you want a more advanced counting system, Professional Blackjack also contains the halves. Blackjack Secrets. This 250-page book explains how to get away with playing a winning game of blackjack in casinos. (You can't win money just by playing

your cards well; you also must avoid drawing attention to your card-counting ability.) It also contains an introduction to the high-low card counting system. Basic Blackjack. This 250-page book is a comprehensive presentation of basic strategy and win rates for all common rules and most exotic rules for the game

350


of blackjack. It also explains methods of getting an edge that do not involve counting cards. Casino Tournament Strategy. This is the book that explains how to get an edge over the other players in tournaments. In 350 pages it covers blackjack, craps, baccarat, keno, and horses. The strategies required for success in a blackjack tournament are completely different from the strategies that are appropriate for blackjack away from tournaments. Professional Video Poker. This is the book that shows how to get an edge on video poker at both Nevada and Atlantic City casinos. The material was developed with the aid of two video-poker pros. (Yes, there are people who support themselves playing video poker.) Optimal Strategy For Pai Gow Poker. This is the book that shows how to get an edge at pai gow poker.. Mike Caro, author of Professional Pai Gow Poker Report, says "Those serious players seeking to make a living at pai gow poker will find Wong's Optimal Strategy for Pai Gow Poker an almost perfect path to profit."

Newsletter Current Blackjack News is a monthly newsletter for blackjack players. It is available electronically, and the' entire content is posted on the bj21.com Internet site. Current Blackjack News is also available by regular mail. When a profitable casino opportunity arises between regular monthly issues, a special issue is published. Special issues have been about such things as casinos paying 2: 1 on naturals or offering early surren-

351 der, blackjack with jokers added to the shoe, and doubleexposure blackjack with 3:2 on naturals.

Internet Web Site On bj21.com you will find stimulating discussions on winning money from casinos, mostly by playing blackjack. Read messages posted by other blackjack players, post your own message if you wish, and chat. There are both free pages and pages requiring payment of a meml>ership fee.

Software Blackjack Count Analyzer. This is a computer program for playing blackjack. It works on IBM PCs and compatible conlputers. Besides playing blackjack, Blackjack Count Afzalyzer generates tables of strategy indexes and runs lligh-speed simulations. It handles all common and uncommon rules variations, and also some rules variations that have never been offered in any casino. A simplified version of it, Blackjack Analyzer, also is available; Blackjack Analyzer uses basic strategy only, whereas Blackjack Count Analyzer supports card counting. Tournament Blackjack. This is a compllter program for playing blackjack tournaments. It works on IBM PCs and compatible computers. It can be used to analyze tournament situations by simulation. Stanford Wong Video Poker. This program, like its predecessor Video Poker Analyzer, is a computer program for playing and analyzing video poker. It works on IBM pes and compatible computers.

352


Also available is VPEXACT, a program whose only function is to cycle through all possible hands of video poker to calculate the payback to the customer.

Ordering Information Write to Pi Vee Press, 7910 Ivanhoe Ave #34, La Jolla, CA 92037-4511 for prices and an order form. Or email [email protected]. Or call (619) 456-4080. Or fax (619) 456-8076.

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