Historical Topics: The Beginnings of Probability Theory

Historical Topics: The Beginnings of Probability TheoryAuthor(s): D. R. GreenSource: Mathematics in School, Vol. 10, No. 1 (Jan., 1981), pp. 6-8Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213600 .

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Historical Top

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Probability Theory by D. R. Green, CAMET, Loughborough University

Bones and Dice in Ancient Times The principal means of introducing the random element into games in classical times was the astragalus - or heel bone. These bones - from various animals - are found in very large numbers and undoubtedly they were used as tally-stones or counters. The shape of the astragalus varies with the type of animal but the more symmetrical ones are the commonest (see Fig. 1). Only four of the six sides are flat enough to permit the astragalus to rest on them.

Fig. 1 Astragali

Sheep Dog

Although used in board games in Egypt as long ago as 3500 BC, it was possibly not much before the birth of Christ that the astragalus was actually used as a die. F. N. David' reports that "A favourite research of the scholars of the Italian Renais- sance was to try to deduce the scoring used". It is widely agreed that the four faces were labelled 1, 3, 4, 6. Because the faces were not equal the empirical probabilities of landing on them were roughly as follows:

Face Probability 1 1/10 3 4/10 4 4/10 6 1/10

A popular game was "rolling of the bones", when four astragali were tossed together. The "best" throw was the "venus" - when each number appears once. The reader might like to confirm that this has a probability of about 384/1000.

What would be the most common throw?2 The more regular die also evolved at about this same period

(perhaps 3000 BC) and examples are found from Iraq, India and Egypt, made of a variety of materials, including pottery.

Divination and Moralising A particular use of the randomising bones or dice, apart from games, was in divination - foretelling the future as is common today in astrology, reading tea leaves, etc. With the spread of

Christianity much of this, but not all, was ultimately swept away.

Cards were not invented until the fourteenth century and even then were very much a luxury so in most gambling situations dice were used until quite modern times. The influence of cards upon the development of Probability theory was thus rather less than that of dice.

In the Middle Ages although gaming with dice was frowned upon it evidently flourished. Indeed, some learned clerics went so far as to invent "moral" games. For example Bishop Wibold of Cambrai around AD 960 enumerated 56 virtues and assigned one to each of the 56 possible ways in which three dice can fall. (The reader might like to check that there are indeed 56 combinations.) In Wibold's game the three dice were thrown, the virtue determined, and the thrower then concentrated on that virtue for some time.

The Beginnings of Probability Theory If a single problem is to be pointed at as encapsulating the germ of Probability Theory it must surely be that found in Luca Paccioli's Summa de arithmetica, geometria, proportioni e pro- portionalitd (1494). The problem is:

"A and B are playing a fair game of balla. They agree to play until one has won six rounds. The game actually has to stop when A has won five rounds and B has won three rounds. How should the stakes be shared out?".

Paccioli argues that the ratio should be 5 : 3 in A's favour but that is not the answer we would accept. The reader might like to calculate the "correct" ratio3. No doubt Paccioli himself saw this as nothing more than a proportion problem but it led to the development of Probability considerations by later mathematicians.

Cardano - famous for his work on Cubic Equations - wrote on many subjects. One of his 242 works was "Liber de Ludo Aleae" (book of games of chance). Being an inveterate gambler himself Cardano was well aware of the empirical probabilities for throwing dice. What he achieved, however, was to set down for the first time ever ideas concerning theoretical probabilities (i.e. abstractions based on equally likely considerations). Referring to the usual six sided die he wrote:

"One half of the total number of faces always represents equality. The chances are equal [i.e. 50-50] that a given face will turn up in three throws, for the total circuit is completed in six. Or again, the chances are equal that one of three given faces turn up in one throw."4

Although possibly written about 1563 this work remained

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unpublished until 1663, some 87 years after Cardano's death. Another early worker in this field was the great Galileo

Galilei (1564-1642). The most well known problem in probability which he tackled was the following:

"When three dice are thrown together there are six combinations which produce a total score of nine:

126 135 144 225 234 333

There are also six combinations which produce a total of 10 (for the reader to check). Why, then, does 10 occur slightly more often than nine?" (A fact spotted by gamblers of the time.)

Galileo realised that the order of the dice mattered, i.e. it is a permutation problem. He enumerated all 216 possibilities (i.e. 6 x 6 x 6) and showed that of these 27 produce a 10 and only 25 produce a 9. Thus this early theorising about probabilities was shown to match up with real experience.

Despite this early work many consider Probability Theory to have been founded by two great seventeenth-century French- men Fermat and Pascal, who corresponded frequently with each other and with other European scholars. The correspond- ence began with the familiar "Problem of Points", an example of which Paccioli had earlier presented but incorrectly solved.

"Two players of equal skill play a game which requires three points to win. Each player lays down 32 gold coins as stake. Suppose the game is unfinished when player A has two points and player B has one point. How should the stake be divided?"

This question really is asking "What is the probability of each player winning, from that position", Paccioli would have argued that A should get twice as much as B, but Fermat and Pascal saw the essence of the problem. In a letter dated 19 July 1654 Pascal wrote to Fermat as follows:

"Now if they play another game and A wins he takes all the stakes, that is 64 gold coins, if B wins it then they have each won two points, and therefore if they wish to stop playing they must each take 32 gold coins.

"Then consider, Sir, if A wins he gets 64 coins, if he loses he gets 32 coins. Thus if they do not wish to risk this last game, A must say: 'I am certain to get 32 coins, even if I lose I still get them. But as for the other 32, perhaps I will get them, perhaps you will get them, the chances are equal. Let us therefore divide these 32 coins in half and give one-half to me as well as my 32 which are mine for sure.'1 A will then have 48 coins and B 16 coins."

Mathematicians on the Continent continued to deal with such problems, and to analyse various gambling games. However in England little interest was shown by the mathematicians, who perhaps disapproved of gambling.

One interesting episode, however, was when Samuel Pepys, the diarist, wrote to Isaac Newton along the following lines:

A elects to throw at least 1 six with 6 throws of a die B elects to throw at least 2 sixes with 12 throws of a die C elects to throw at least 3 sixes with 18 throws of a die. Who has the easiest task, or are they all the same?

Realising that Newton might not approve of the bet which Pepys had made with someone over this, he wrote to Newton saying that a "Mr Smith" wanted to know the answer "due to his soe earnest an application after truth".

Newton soon solved the problem. For A the chances are 31031/46656 (=0.665) and for B 1346704211/2176782336 (= 0.619). Not surprisingly Newton didn't do the numerical work for C! However, he said it would be a lower chance, and ponted out the method.

There are two rather different ways to look upon probability (actually there are more than two - see Buxtons for an illumi-

nating discussion). The first is a mathematical definition, based on equally likely outcomes, something along the following lines:

"If an event can happen in n different ways, all of which are

equally likely, and if s of these are defined as 'success', the

probability of 'success' is S

The second concept is a statistical or experimental definition, based on frequency ratios, as follows:

"If n trials of an experiment are conducted and of these s have been designated 'successful' then the probability of success is S ,,

n These may appear similar but this is not so. The first defini-

tion is an abstract theoretical one, based upon a mathematical model of reality. The second is a purely practical one, based on actual experience.

Down the centuries both these ideas were recognised but the precise relationship between them was unclear. The funda- mentally important connection was first enunciated by a mem- ber of that extraordinary family of mathematical geniuses - the Bernoullis.

In his Ars Conjectandi, published posthumously in 1713, James Bernoulli set out his theorem, which is often called the "Law of Large Numbers". Put simply it says:

"Statistical probability can be brought as close to mathematical probability, with as high a degree of certainty as we wish, by repeating the experiment a sufficient number of times."

To show more clearly what Bernoulli wrote we will look at his illustrative problem. Consider a bag containing 50 balls, 30 coloured white and 20 coloured black. When a single ball is picked out at random, then replaced, and the experiment repeated many times, the results usually settle down to around 60% white (and 40% black). Generally it will not be exactly 60% white, of course, but fairly close. How many trials are needed in order to be 99% sure of getting between 55% and 65% white balls? (In other words for the statistical ratio is to be within

- 5% of the mathematical ratio.) The mathematical probability model for this situation is

(0.6 + 0.4)" where there are n trials of the experiment. By expanding this, using the Binomial expansion, we get:

(0.6)" probability of n whites and 0 blacks

+(n)(0.6)"-1 (0.4)1 probability of n-1 whites and 1 black

+-2 (0.4), probability of n - 2 whites and 2 blacks

+ (0.4y probability of 0 whites and n blacks

It is, therefore, possible to work out all the probabilities. For example if n = 20 then if 55% to 65% of the trials are to

result in a white ball being selected this means 11, 12, or 13 white balls out of the 20 selections. The probability of this happening is:

20)(0.6)13 (0.4)7+ 20)(0.6)12 (0.4)8+ (20)(0.6)" (0.4)9 = 0.15974+0.17971 + 0.14563 = 0.4851

Thus about 481/2% of all such experiments should result in 55% to 65% of white balls being selected.

What happens as n is increased? Table 1A shows the effect of increasing the number of trials in an experiment, and it illustrates the kind of pattern which Bernoulli discovered.

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Table 1 A B

Trials, n 55% to 65% Prob- 0.6 n- 1 to Prob- of n ability 0.6 n+ 1 ability

20 11 to 13 0.485 11 to 13 0.485 40 22 to 26 0.580 23 to 25 0.371 60 33 to 39 0.644 35 to 37 0.307 80 44 to 52 0.696 47 to 49 0.268

100 55 to 65 0.739 59 to 61 0.240 200 110to 130 0.871 119to 121 0.171 500 275 to 325 0.932 299 to 301 0.109 700 385 to 455 0.994 419 to 421 0.092

1000 550 to 650 0.999 599 to 601 0.077

Note that as n increases the probability of picking between 55% and 65% of white balls increases. By n= 700 we have a 0.99 probability (or 99% certainty) of being within those limits. This illustrates the law which Bernoulli discovered - if enough trials are taken then you can be as certain as you like of getting as close as you like to the theoretical proportion (in our case 60%).

Note carefully that Bernoulli's Theorem is to do with pro- portions. Once you restrict yourself to actual numbers things are very different. The probability of being within L 1 ball of

the 60% white decreases with increasing n, as can be shown by reference to the Table lB.

Bernoulli treated this problem in a general form by expanding (r+s)(r+s), and then used as his illustrative example the one we have just considered6. Bernoulli introduced his theorem in his book as follows:

"This is therefore the problem that I now want to publish here, having considered it closely for 20 years, and it is a problem of which the novelty, as well as the high utility, together with its grave difficulty, exceed in weight and value all the remaining chapters of my doctrine." (Ars Conjectandi: Pars Quarta.)

With this important theorem established the subject of Probability was well on the road.

Notes and references

1. David, F. N. (1962) Games, Gods and Gambling, Griffin. 2. Two threes and two fours. 3. 7:1. 4. Ore, 0. (1953) (translated by S. H. Gould) "Liber de Ludo Aleae",

Cardano the Gambling Scholar. 5. Buxton, J. R. (1969 and 1970) "Probability and its measurement", Mathe-

matics Teaching, 49, 4-12 and 50, 55-61. 6. See Chapter 13 of Reference 1.

Negative digits (or tricimals)

by John T. S. Mills, Department of Science Education, University of Warwick

I was most interested to read John Farina's article "Negative Digits"', as I have used this notation on occasion as a novel arithmetic system. I would like to suggest a possible teaching approach which does not involve the use of sequences of powers of 3 but uses this notational system to look at some of the important properties of a number system.

What is the minimum set of axioms (assumptions) we need to make in order to deduce all the basic arithmetic of this system? From the practical situation of weights of 1 and 3 units we abstract the notation 1 + 1 = 1f and the further fact (1 + 1)+ 1 = 10. Using place value this yields the deduction T + 1 = 0 and also 1+ T = 0 either using associativity or postu- lating commutativity. Is it possible to deduce that this is the familiar 0 with the property x + 0 = x or is it necessary to introduce another axiom? I leave the answering of this question as an exercise to the reader. Assuming that we now have 0 as the zero it is possible to prove that T+ T=1T1 by using associativity as:

((1+ 1)+ 1)+ 1=(1+ 1)+(1+ 1)

Having produced the basic rules of arithmetic manipulation, addition and multiplication blocks are worth constructing to show the symmetry of this notational system.

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+ T 0 1 x 1 0 1

01 T 1 0 0 0 1

1 0 1 11 1 1 0 1

In practice multiplication in trinary usually turns out to be simpler than binary, there being fewer carries due to 1, 1 elimination.

Division is, I agree, awkward and this has led me into an investigation of multiplicative inverse and hence "tricimal" expansions.

In the usual notation we have:

= 0.1 1

= 0.011 10 100

What of 1- )? As a first attempt one may solve xx 1I=1 by successive trials modifying the next tricimal place according to whether the current product is greater or less than one.

11 x 0.1 =1.1 1Tx0.11 =i.T+.1T =1.01 1Tx O.111 = 1.0+ .0T= 1.00T

The pattern being established we have for ( I1 =0.1111 ... 11



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Historical Topics: The Beginnings of Probability Theory