Chance and Choice by Cardpack and Chessboard: An Introduction to Probability in Practice by Visual Aids: Volume I

CHANCE AND CHOICE BY CARDPACK AND CHESSBOARD AN INTRODUCTION TO PROBABILITY PRACTICE BY VISUAL AIDS VOLUME

IN

1

by

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1

1

exalt the

>h"Lild

seems

1

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-

N

po\M

i

a

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1

is

seek

lather

is

[

II

I'

nf

out

N

I

V i, R

the

>

1

iiuin

V

I

i:




and

the left

If

and

right in accordance with the Cartesian convention, a a and b When we wish to signify that -f b.

so that in general

+ >




x

is

4,

numerically-

<


One

other symbol is of great importance, since so venient approximations precise enough for practical needs. to

y.

many

statistical

formulae are con-

In contradistinction to =, our

~

signifies approximately equal to. reasonable to assume that some readers with sufficient background to benefit from any uses this book may have will have forgotten some of the algebra, more especially differential calculus, necessary for the exposition of principles set forth in later chapters. Accord-

symbol

It is

ingly,

the writer has taken the precaution to intercalate sections to provide opportunities of who have not ready access to text-books of mathematics. The exercises

revision for readers

are designed, where possible, to anticipate subsequent themes and an asterisk signifies that a result will be of use at a later stage. The insertion of certain lemmas as exercises on a relevant class of operations is intentional to dispense with the need for digressions which distract attention ;

from the main

issue

;

and the reader

be well advised to pay attention to them, as they

will

arise.

SYMBOLISM OF CONTINUED ADDITION AND MULTIPLICATION

1.01

In the mathematical analysis of choice and chance we frequently have recourse to expressions It is customary to represent the operation of summation involving continued sums or products. briefly by use of the Greek capital s (sigma) thus :

x

tfo

+

+

i

In operations involving summation,



6

.

.

5

'Another useful formula involving factors of a product

.

-=

4

(7

factorial

.

6

.

5

4

.

powers

.

3

.

2

.

^

1)

(3

2

.

.

1).

deducible by reversing the order of the

is

:

n(n If

we

+

l)(n

+ 2)

.

.

r factors

.

= n(n +

.

.

+ r - 2)(n + r -

(n

.

+ r - !)( + r - 2) + 2)(n + l)n __ + r l)(n + r +r-l 2) ... (w + r~l - r + !)( r factors. + r - l)(n + r - 2) - l) /. n(n f l)(n + r factors 2) (n + r (n

(

+ 2)

1).

we have

reverse the order,

= = (n

l)(n

.

.

.

(

1

.

.

1)

.

.

.

(f)

--=

.

.

.

.

(v)

Several important properties of factorial powers follow from the following relation which holds

good

if

x

n

(n}

>m

:

= x(x -

= [x(x =#

(rn) .

- 2)

l)(x

(#

1)

.

.

.

w)

.

.

.

(x

-m (x

(n ~

__

- m + 2)(x

4- !)][(*

-

-m + l)(x - m)(x - m -

1)

.

.

.

(x-n+ 2)(x - + if

-mm)(x

1)

.

.

1)

.

m

^r 2)(#

n

m +

1)]

m) .

Subject to the same condition, the laws of composition of factorial indices follow at once from this, viz.

:

(n)

^

[

X (m)]2( x

- m )

3!

Formulae for the series of higher dimensions can be severally obtained by the method of 1.05 below, but the generalisation of the pattern common to the above is justifiable by induction. For the first four series we have

CHANCE AND CHOICE BY CARDPACK AND CHESSBOARD

16

= (n + 0! in accordance with (viii) of 1.01. 1) - = ^T+7 - !> -Ml. I/?, = (n + 2 - 1)' 2!. B = Fn +3

opn

-=-

(0>

1

2>

-T-

-f-

3

(

In general,

we have

In accordance with (iv) in 1.01

Thus, the 4th term of the series ..,

From

(i) it

*"

1,

we may

7,

also write this as

28 ...

(6+4-1)!

etc., in

9.8.7 "3.2.1

9! "

6! 3!

61(4-1)!

agreement with the foregoing table

= 84.

follows that l

rf!

(1-1)1

d\

+ 2) - 1)1 (d + 1 + 2 ,(*

-F. rs

-

2

+ 1)! 2! (J++3-1)! + !)! (3 -1)1' ..._. (d+ 1 + 1)1 WJ Hrf + 1M fa - lil" (
the appropriate symbol for the series whose generative A.P. sional A.P.

=

=

+

numbers is *Fn The latter are the only ones we shall make use of in what follows ; a helpful proficiency exercise to explore the properties of others. With the help of formula Fig. 7, the student should be able to establish by induction and to check the more general of which (ii) above is a particular case :

is

the natural

but

.

it is

(n

+ d-2) (

* (\Fn ).

I

>2

k

x~n

2*

2 ;

for x

n

3 ;

*n

1

Z

;

(n) alone for

x~n

x**n

Z**;

Z*

*

1

Z

n

8 -*?

Show

that

.

>

3p.

6* 4

>

5

n

Z

*';

Find the numerical values of 2

10.

\Fn 6*n

i

Obtain an expression involving one unknown

Z*

9.

F

3^6

>

that

*=n

8.

6

/r

3^6

10 members of the series

X

5.*

4

3 /r

2 7T >

4*n

4.*

numbers and the 4th -dimensional

12 triangular numbers, the tetrahedral

class.

i/r

3.

1.02

4p. e-^S

>

1

*

-

;

CHANCE AND CHOICE BY CARDPACK AND CHESSBOARD

IP FIG.

9.

One way

I

+70,..)*

of finding a Formula for the Figurate

:R

:R

Fio. 10.

1

7

'.,

Numbers

*

4

of Fig.

8.

$..

-

*

Another way of finding a Formula

for the Figurate

Numbers

of Fig.

8.

FIGURATE NUMBERS AND FUNDAMENTAL APPROXIMATIONS

25

Find an expression for

11.

|Fn+1

(a)

tFn . x+l

(b)

;

.

Find the numerical values of

12.

x-

10

x-$

*-12

*-ll

*-10

*-12

*-ll

*-5

*-3

*-4

^-6

x-7

Evaluate

13.

*-3

^

-

x-3

2

Devise a formula for

14.*

x-b

*-6

Repeat No. 5 of Ex. 1.01 with due regard

15.

to the

PASCAL'S

1.03

meaning of

s

Fn

when

s

>

3.

TRIANGLE

We may set out the series generated successively by the units, natural numbers and triIn such an arrangement there is angular numbers of 1.02 in successive columns, as in Fig. 11. a row of terms corresponding to those of each column. By sliding successive columns downwards through

though (r

+

1)

rows we get the arrangement (Fig. 12) known as Pascal's triangle, from the time of Omar Khayydm. In this arrangement the rth row has terms whose rank we label from to c> so that the cth column starts at the end of the cth 1, 2, 3,

it

...

etc.,

dates at least

row.

We

shall label a Pascal

reference to Figs. 11 and 12 2(0)

2 (1) 2,

number r(c) we see that

= = -F = ^2-