Maurice G. Kendall - Studies in the history of probability and statistics V

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Studies in the History of Probability and Statistics. V. A Note on Playing CardsAuthor(s): M. G. KendallSource: Biometrika, Vol. 44, No. 1/2 (Jun., 1957), pp. 260-262Published by: Biometrika Trust

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260

Miscellanea

raised to the

199th power using tables of 15

figure logarithms.

The first term

in

equation (2)

was

obtainable without quadrature,

using the same tables.

REFERENCES

DAVID,

H. A.,

HARTLEY,

H. 0. &

PEARSON,

E. S. (1954). Biornetrika, 41, 482-93.

NATIONAL

BUREAU OF STANDARDS

(1953).

Tables of

Normal

Probability Functions.

Applied Mathe-

matics

Series, no. 23. Washington: U.S.

Department of Commerce.

PEARSON,

E.

S.

(1932).

Biometrika, 24, 404-17.

PEARSON,

E. S. &

HARTLEY,

H.

0. (1942). Biometrika, 32, 301-10.

PEARSON,

E. S. &

HARTLEY,H. 0. (1954).

Biometrika Tablesfor Statisticians, 1.

Cambridge University

Press.

RUBEN,

H.

(1954). Biometrika, 41, 200-27.

TIPPETT,

L.

H. C.

(1925). Biometrika, 17, 364-87.

'TUKEY, . W.

(1955). Biometrika, 42, 480-5.

Studies in the history of probability and statistics. V. A note on playing cards

BY M. G. KENDALL

Research Techniques Unit, London School

of Economics

1.

In

an

earlier article in this series (Kendall, 1956) I referred

briefly to the introduction

of

playing

cards into

Europe.

Subsequent correspondence arising out of that reference suggests that it may be

useful

to

expand

a

little

what was there said about the impact of cards on gambling.

2.

Playing cards

as we

know them to-day in western Europe can be traced back in a clear line of descent

to

the

beginning

of the

fifteenth century; but from that point

backwards their history

becomes

more

and

more

vague

and

their genealogy more and more fabulous.

Where they originated is

unknown.

Claims

have been

put

forward

on behalf of origins in China, India,

Arabia and Egypt. It

is

at

least

equally

possible that they were independently invented in Europe. From the first the pictorial representations

on the cards

were

thoroughly Western and do not suggest, to my eye

at least, any trace of Eastern descent

such,

for

example, as does

the rook in chess.

*

Gambling with paper tickets is said to have been known in

China

in

the

twelfth

century,

and

it is possible that the idea of playing cards drifted across

to

Europe

along

one of the

early trade routes; but for the translation of the

idea into practice one does not need to

look

further afield than

fourteenth-century Italy.

3. No

mention of

cards

has been

traced in the West before

A.D.

1350, and the absence of reference

in

authors

like

Chaucer and Dante, who mentioned everything, shows

that they cannot have been known

much before that

date. They seem to have spread in the

Mediterranean countries and Germany fairly

rapidly. Numerals are

said to have been added to the picture cards at Venice in

A.D.

1377. Cards are

mentioned

at

Niurnberg

n

A.D. 1380. In A.D. 1397 there appeared an ordinance in Paris prohibiting play

at

various

games, including cards, to that part of the population

who were engaged in manufacturing.

(The story that cards were invented to amuse the mad Charles VI of France is false, although packs were

made for him

in

A.D.

1393.) Specimens of such early dates have

not survived. By A.D. 1423, however,

the

card

pack appears

to

have

evolved into its modern form. San

Bernardino, whose sermon served

me

in

good

stead in the

previous article (1956), refers to charticellae in

quibus

variaefigurae

pingantur,

and

goes

on to mention

the four

suits, the Kings, Queens, Valets and Chevaliers and, as I interpret him,

the

trump

cards of the

tarot

pack.t

*

Modern

tarot

packs are

worthless evidence in this connexion.

Under the influence

of

occultists,

notably Court

de

Gebelin,

who suggested in the

eighteenth century that the trump cards

incorporated

the

lost book

of

Thoth,

and

Eliphas

Levi,

whose work on magic

popularized the idea in the nineteenth

century, tarot cards have

acquired symbols such as sphinxes which are

absent from earlier western

cards.

f

The

early

suits were

clubs,

coins

(diamonds),

cups (hearts)

and

swords

(spades),

which

San

Ber-

nardino

identifies

with

brutality, avarice, drunkenness and hatred. The

court cards denote those

who

are outstanding in these vices. 'Presbyterorum et presbyterarum in numerabilem multitudinem esse

volo: unde admittantur

pusilli et magni, et

feminae et masculi, periti et ignari, sapientes et

stulti.'

A hundred

and

fifty years later one John

Northbrooke was inspired by this passage to coin the

famous

description

of cards

as 'the devil's picture

books'.

Miss

Gertrude Moakley (1956) has recently

suggested that the trump

cards were representations

of

the

Trionfi

described

by

Petrarch

in

one of his

most popular poems.

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Miscellanea 261

4. The game of tarock (French tarot,

Italian

tarocchi)

is still played in

southem

Europe in various

versions and is probably the oldest card

game. In its modern form the tarot pack consists of 56 ordinary

cards divided into four suits, 21 trump cards and a wild card or joker, 78 cards in

all. Fifteenth-century

packs of this type exist. There are some

early packs of even larger size, notably the Minchiate pack of

97 cards, 56 ordinary cards, 35 trump

cards and 6 wild cards. Historians of playing cards also mention

a very rare set of engravings known as the Mantegna tarot; but they are probably not by Mantegna,

and I doubt very much whether they

are tarot cards. They are engraved on sizeable but thin sheets of

paper, and

can

hardly have been used

in any sort of game involving shuffling

and dealing; they are

divided into five sets of ten, the first, for

example, enumerating various social grades from the beggar to

the Pope, the second giving the nine Muses and Apollo, the third giving ten branches

of learning, and so

on. My

own

opinion is that these cards were a teaching device and that the unknown

inventor of the

tarot

pack copied some of them. Where

he got the others is a mystery unless Miss Moakley is right in

identifying them with Petrarch's

Trionfi.

5. At some unknown point of time the tarot pack was simplified, or so I believe.

The trump cards.

were dropped, and of the 56 ordinary cards one of the court cards was also dropped.

(In most countries

it was the Chevalier who was dropped,

but in Spain it was the Queen, for reasons which it is interesting

but unprofitable to speculate upon.) Thus there evolved the basic pack of 52 cards

which is in general

use to-day. The tarot pack survived independently, but in northern Europe is now mainly used for

fortune-telling.

6.

The

student

of

the

history

of

probability

is interested

in

these

matters

only

in

two

respects:

the

degree

to which cards extended and encouraged gambling,

and

the

reasons for the choice

of

the number

of cards in the early tarot packs. From what San Bernardino says it seems

that

gambling

began at

a

very

early stage; presumably, as soon as a

game came into existence, the adversaries

began to wager on the

outcome. However, extensive gambling with cards was of very slow development.

Cardano mentions

the game of primero, but early writers

on chance confine themselves mainly to dice. The reasons,

I

think,

were

twofold: first, the permutational

arithmetic required to deal with probabilities at cards

was too

complicated; secondly, cards

were

very expensive

and dice much more

common.

Cards

did

not oust

dice until

the eighteenth century.

A third reason, possibly, is that cards and backgammon involved

more

skill

and

had a

higher social status. James

I

(1603) puts it rather

well:

'As for sitting, or human pastimes-since they may at times supply the room which, being empty,

would be patent to pernicious idleness-I will not therefore agree

with the curiosity of some

learned men

of

our age

in

forbidding cards,

dice and such like games of hazard;

when it is

foul

and

stormy weather,

then I

say, may ye lawfully play

at the cards or

tables; for,

as

to

dicing,

I think it

becometh

best

deboshed

soldiers to play at on the heads of their drums, being only ruled by hazard,

and subject to

knavish

cogging;

and

as

for

the

chess,

I

think

it over-fond because

it is overwise and

philosophic

a

folly.'

James, apparently,

was not

very good

at chess, but

his

balanced

broadmindedness

in

a

Puritan

age

compels respect.

7.

It

is

interesting

to

inquire why

the early tarot packs

consisted of 56

+

21

+

1

or 56

+

35

+

6

cards,

but

before dabbling

in

numerology

let us note how treacherous

a

subject

it is.

One

can

make

out

a

very

good

case for a

connexion between

the modern

pack

of

playing

cards

and the calendar.

The

four

suits

correspond

to

the

four

seasons,

the thirteen cards

in

a

suit

correspond

to the

lunar

months,

the

52 cards

to the weeks of the

year.

If we score 11 for a Knave, 12 for a Queen, and 13 for a King, and add all the

points

for

the

52 cards we

get 364,

which, adding

one

for

the

joker, gives

us

the number

of

days

in the

year.* Many

a

historical

point

has been

argued

on less coincidental

evidence than

this,

but there

is,

in

fact,

no

connexion

between

the playing pack

and the calendar. The

early

suit

cards were

56 in number.

8.

Nevertheless,

there are some

striking

resemblances between

the

number

of the cards

in

the

early

packs and the number of ways

in

which dice can

fall.

The

main

number, 56,

is

the number of

ways

in

which

three

dice

can be thrown, permutations excluded,

and

I

pointed

out

in

my previous

article

(1956)

that

these

ways

were well

known

by

the fourteenth century.

The

number

21, likewise,

is the

number

of

ways

in which two

dice can be thrown,

permutations excluded. The number 35

of

the larger

Minchiate

tarot could

arise

either

as

the

number

of

ways

of

throwing

three dice

when

sixes

are

ignored,

or the

number

of ways

of

throwing four (four-sided)

astragali. I am not prepared to lean very heavily

on these

coincidences;

but

they suggest

that

perhaps

the constructors

of the first

packs, having

to choose

somehow,

were

influenced

by

their

knowledge

of

dice-throwing.

*

Nor is this the worst. The number

of

letters

in

the sequence Ace, Two, Three,

...,

ITen,

Jack, Queen,

King, is 52; so also in the sequence As, Deux, Trois, .

.,

Dix, Valet, Reine, Roi and in As, Zwei, Drei,

..

.

Zehn, Bube, Dame, Konig, the ch being taken as one letter. I owe this information to the firm

of

Thos.

de

la

Rue and Co. Ltd.









262

Miscellanea

9. In conclusion, I should like to correct one

guess

made

in

my

earlier article.

I

suggested

that

the

game

of hazard

was brought back to Europe by

the third crusaders. It may, indeed, have been

brought

back in some such way, but if so, must have been

imported by earlier crusaders.

The word hcwartoccurs

in line 10557 of Wace's Le Roman De Brut, dated

A.D.

1155, and also in Chr6tien de Troyes'

Ereo et

Enide,

line 356, dated

A.D.

1160-70. For these

references I

am

indebted

to Prof. Brian

Woledge,

who remarks, incidentally, that the appearance of the initial 'h' in 'hazard'

is

an etymological

mystery which has never been solved.

REFERENCES

JAms

I (1603).

Bawilikon

Doron, or a

King'8

ahristian Duty

toward89God.

KENDALL,

M. G. (1956). Studies in the history of probability and statistics. II. The beginnings

of

a calculus of probabilities. Biometrika, 43, 1.

MOAKTEY,GERTRUDE

(1956). The tarot trumps and Petrarch's

Trionfi.

Bull.

N.Y. Publ.

Lib.

60,

55.

A

singularity

in the estimation of binomial variance

By ALAN STUART

Research

Technique8

Unit, London School of Economics

SUMMARY.

For the

symmetrical

binomial distribution, the limit distribution

of the

sample

variance is non-nornal and has variance of order

I/n2.

1. For the binomial distribution, the sample proportion of

'successes',

p, is a sufficient estimator of

the

probability parameter 7T, nd

y

=

p(l-p) n/(n-1) (1)

is an unbiased estimator of

7T(I

-

7T),

which is n times the sampling variance

of p, where n is sample size.

p(l -p)

is

the sample variance. Since

y is a function of the sufficient statistic, it is the minimum variance

unbiased estimator of its expectation. Its sampling variance is

V(y)

=

(:

1){V(p)+ V(p2)-2C(p,p2)},

which is

expressible

in

terms

of the moments about the origin of p as

V(y)

=

I

3

2 2 1

On

substitution

for these moments

we

find

V(y)

= ((l

)(- r2)2+2

+T( }

).

(2)

nnI

If

77 i

(2)

gives

V

7(y

I

7T

i) -7T(

-

7T) (I

-

27T)2

n,

(3)

while if

iT

=

i

(2)

becomes

V(y

1

7T

=

i)

=

I

/{8n(n- 1)}.

(4)

It

is

easily

confirmed that the

right-hand

sides

of

(3)

and

(4)

are the information bounds to the

sampling

variance

of an estimator of

7T(

1-

7T),

to

order

n-1,

n-2

respectively

in

the two cases.

The fact

that

at

fT

=

i

the variance

of

y

is

of lower

order than

n-I raises

the

question

whether

its

limiting distribution is normal

in

that

case. In the following sections it is shown that this is not so.

2. The characteristic function

of

p(l -p) may

be

written

=

E{exp

[o(p

-

p2)]}

=

exp (0fT2)

E{exp [9p(l

-

2rT)] xp

[-

O(p

-

7T)2]},

(5)

where 0 = it.

Since,

for random variables u, v,

E(uv)

=

E(u) E(v) + (u, v),

we

may replace

the

expectation

of the two

exponential

terms

in

(5) by

the

product

of their

expectations

plus their

covariance

C =

C{exp[Op(l-2r2)],

exp[-O(p-7T)2]}.

(6)







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Maurice G. Kendall - Studies in the history of probability and statistics V