Classic Problems of Probability E-Book

Prakash Gorroochurn

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 469

Veröffentlichungsjahr: 2012

Beschreibung

Winner of the 2012 PROSE Award for Mathematics from The American Publishers Awards for Professional and Scholarly Excellence. "A great book, one that I will certainly add to my personal library." --Paul J. Nahin, Professor Emeritus of Electrical Engineering, University of New Hampshire Classic Problems of Probability presents a lively account of the most intriguing aspects of statistics. The book features a large collection of more than thirty classic probability problems which have been carefully selected for their interesting history, the way they have shaped the field, and their counterintuitive nature. From Cardano's 1564 Games of Chance to Jacob Bernoulli's 1713 Golden Theorem to Parrondo's 1996 Perplexing Paradox, the book clearly outlines the puzzles and problems of probability, interweaving the discussion with rich historical detail and the story of how the mathematicians involved arrived at their solutions. Each problem is given an in-depth treatment, including detailed and rigorous mathematical proofs as needed. Some of the fascinating topics discussed by the author include: * Buffon's Needle problem and its ingenious treatment by Joseph Barbier, culminating into a discussion of invariance * Various paradoxes raised by Joseph Bertrand * Classic problems in decision theory, including Pascal's Wager, Kraitchik's Neckties, and Newcomb's problem * The Bayesian paradigm and various philosophies of probability * Coverage of both elementary and more complex problems, including the Chevalier de Méré problems, Fisher and the lady testing tea, the birthday problem and its various extensions, and the Borel-Kolmogorov paradox Classic Problems of Probability is an eye-opening, one-of-a-kind reference for researchers and professionals interested in the history of probability and the varied problem-solving strategies employed throughout the ages. The book also serves as an insightful supplement for courses on mathematical probability and introductory probability and statistics at the undergraduate level.

Leseprobe

Contents

Cover

Title Page

Copyright

Dedication

Preface

Acknowledgments

Problem 1: Cardano and Games of Chance (1564)

1.1 Discussion

Problem 2: Galileo and A Discovery Concerning Dice (1620)

2.1 Discussion

Problem 3: The Chevalier de Méré Problem I: The Problem of Dice (1654)

3.1 Discussion

Problem 4: The Chevalier de Méré Problem II: The Problem of Points (1654)

4.1 Discussion

Problem 5: Huygens and the Gambler's Ruin (1657)

5.1 Discussion

Problem 6: The Pepys–Newton Connection (1693)

6.1 Discussion

Problem 7: Rencontres with Montmort (1708)

7.1 Discussion

Problem 8: Jacob Bernoulli and His Golden Theorem (1713)

8.1 Discussion

Problem 9: De Moivre's Problem (1730)

9.1 Discussion

Problem 10: De Moivre, Gauss, and the Normal Curve (1730, 1809)

10.1 Discussion

Problem 11: Daniel Bernoulli and the St: Petersburg Problem (1738)

11.1 Discussion

Problem 12: D'Alembert and the “Croix ou Pile” Article (1754)

12.1 Discussion

Problem 13: D'Alembert and the Gambler's Fallacy (1761)

13.1 Discussion

Problem 14: Bayes, Laplace, and Philosophies of Probability (1764, 1774)

14.1 Discussion

Problem 15: Leibniz's Error (1768)

15.1 Discussion

Problem 16: The Buffon Needle Problem (1777)

16.1 Discussion

Problem 17: Bertrand's Ballot Problem (1887)

17.1 Discussion

Problem 18: Bertrand's Strange Three Boxes (1889)

18.1 Discussion

Problem 19: Bertrand's Chords (1889)

19.1 Discussion

Problem 20: Three Coins and A Puzzle from Galton (1894)

20.1 Discussion

Problem 21: Lewis Carroll's Pillow Problem No: 72 (1894)

21.1 Discussion

Problem 22: Borel and A Different Kind of Normality (1909)

22.1 Discussion

Problem 23: Borel's Paradox and Kolmogorov's Axioms (1909, 1933)

23.1 Discussion

Problem 24: Of Borel, Monkeys, and the New Creationism (1913)

24.1 Discussion

Problem 25: Kraitchik's Neckties and Newcomb's Problem (1930, 1960)

25.1 Discussion

Problem 26: Fisher and the Lady Tasting Tea (1935)

26.1 Discussion

Problem 27: Benford and the Peculiar Behavior of the First Significant Digit (1938)

27.1 Discussion

Problem 28: Coinciding Birthdays (1939)

28.1 Discussion

Problem 29: Lévy and the Arc Sine Law (1939)

29.1 Discussion

Problem 30: Simpson's Paradox (1951)

30.1 Discussion

Problem 31: Gamow, Stern, and Elevators (1958)

31.1 Discussion

Problem 32: Monty Hall, Cars, and Goats (1975)

32.1 Discussion

Problem 33: Parrondo's Perplexing Paradox (1996)

33.1 Discussion

Bibliography

Photo Credits

Index

Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved

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Library of Congress Cataloging-in-Publication Data:

Gorroochurn, Prakash, 1971-

Classic problems of probability / Prakash Gorroochurn.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-118-06325-5 (pbk.)

1. Probabilities–Famous problems. 2. Probabilities–History. I. Title.

QA273.A4G67 2012

519.2–dc23

2011051089

To Nishi and Premal

operae pretium est

Preface

Having taught probability for the last 12 years or so, I finally decided it was time to put down my “experiences” with the subject in the form of a book. However, there are already so many excellent texts in probability1 out there that writing yet another one was not something that really interested me. Therefore, I decided to write something that was a bit different.

Probability is a most fascinating and unique subject. However, one of its uniquenesses lies in the way common sense and intuition often fail when applied to apparently simple problems. The primary objective of this book is to examine some of the “classic” problems of probability that stand out, either because they have contributed to the field, or because they have been of historical significance. I also include in this definition problems that are of a counterintuitive nature. Not all the “classic” problems are old: Problem 33: Parrondo's Perplexing Paradox, for example, was discovered as recently as 1996. The book has considerable coverage of the history of the probability, although it is not a book on the history of the subject. The approach I have adopted here is to try to offer insights into the subject through its rich history. This book is targeted primarily to readers who have had at least a basic course in probability. Readers in the history of probability might also find it useful.

I have worked hard to make the presentation as clear as possible so that the book can be accessible to a wide audience. However, I have also endeavored to treat each problem in considerable depth and have provided mathematical proofs where needed. Thus, in the discussion of Problem 16: The Buffon Needle Problem, the reader will find much more than the conventional discussion found in most textbooks. I discuss alternative proofs by Joseph Barbier, which lead to more profound and general results. I also discuss the choice of random variables for which a uniform distribution is possible, which then naturally leads to a discussion on invariance. Likewise, the discussion of Problem 19: Bertrand's Chords involves much more than stating there are three well-known possible solutions to the problem. I discuss the implications of the indeterminacy of the problem, as well as the contributions made by Henri Poincaré and Edwin Jaynes. The same can be said of most of the problems discussed in the book. The reader will also find treatments of the limit and central theorems of probability. My hope is that the historical approach I have adopted will make these often misunderstood aspects of probability clearer.

Most of the problems are more or less of an elementary nature, although some are less so. The reader is urged to refrain from focusing solely on the problems and their solutions, as it is in the discussion that the “meaty” parts will be found. Moreover, the selection of problems here are not necessarily the most important problems in the field of probability. Although a few such as Problem 8: Jacob Bernoulli and his Golden Theorem, Problem 10: de Moivre, Gauss, and the Normal Curve, and Problem 14: Bayes, Laplace, and Philosophies of Probability are important, many others are more interesting than they have been decisive to the field. Thus, few readers who are not very conversant with the history of probability are probably aware that scientists like Galileo and Newton ever used probability in their writings. Nor would anybody contend that these two men made any fundamental contributions to the theory of probability. Yet, I am hopeful that the manner in which these scientists tackled probability problems will awaken some interest in the reader.

I have refrained from giving extensive biographies of the mathematicians discussed in this book because my focus is mainly on the problems they solved and how they solved them. Readers eager to know more about these scientists should consult the excellent Dictionary of Scientific Biography, edited by C.-C. Gillispie.

If I had a criticism of the book, it would be the unevenness in the treatment of the problems. Thus, the reader will find more than 20 pages devoted to Problem 4: The Chevalier de Méré Problem II: The Problem of Points, but less than 5 pages spent on Problem 15: Leibniz's Error. This is inevitable given the historical contexts and implications of the problems. Nonetheless, I would be remiss to claim that there was no element of subjectivity in my allocation of space to the problems. Indeed, the amount of material partly reflects my personal preferences, and the reader will surely find some of the problems I treated briefly deserved more, or vice versa. There is also some unevenness in the level of difficulty of the problems since they are arranged in chronological order. For example, Problem 23: Borel's Paradox and Kolmogorov's Axioms is harder than many of the earlier and later problems because it contains some measure-theoretic concepts. However, I also wanted the reader to grasp the significance of Kolmogorov's work, and that would have been almost impossible to do without some measure theory.

Most of the citations were obtained from the original works. All the French translations were done by me, unless otherwise indicated. Although this book is about classic problems, I have also tried to include as many classic citations and references as possible.

I hope the book will appeal to both students, those interested in the history of probability, and all those who may be captivated by the charms of probability.

Prakash [email protected] 2012

Note

1. For example, Paolella (2006, 2007), Feller (1968), Ross (1997), Ross and Pekoz (2006), Kelly (1994), Ash (2008), Schwarzlander (2011), Gut (2005), Brémaud (2009), and Foata and Fuchs (1998).

Acknowledgments

My foremost thanks go to my friend and colleague Bruce Levin for carefully reading an almost complete version of the manuscript and for making several excellent suggestions. Bruce has been a constant source of encouragement, motivation, and inspiration, and I hope these few words can do justice to him. Arindam Roy Choudhury also read several chapters from an earlier version, and Bill Stewart discussed parts of Problem 25 with me. In addition, I am grateful to the following for reading different chapters from the book: Nicholas H. Bingham, Steven J. Brams, Ronald Christensen, Joseph C. Gardiner, Donald A. Gillies, Michel Henry, Collin Howson, Davar Khoshnevisan, D. Marc Kilgour, Peter Killeen, Peter M. Lee, Paul J. Nahin, Raymond Nickerson, Roger Pinkham, and Sandy L. Zabell. I would also like to thank the following for help with the proofreading: Amy Armento, Jonathan Diah, Guqian Du, Rebecca Gross, Tianxiao Huang, Wei-Ti Huang, Tsun-Fang Hsueh, Annie Lee, Yi-Chien Lee, Keletso Makofane, Stephen Mooney, Jessica Overbey, Bing Pei, Lynn Petukhova, Nicolas Rouse, John Spisack, Lisa Strug, and Gary Yu. Finally, I thank the anonymous reviewers for their helpful suggestions.

My thanks also go to Susanne Steitz-Filler, editor at Wiley. Susanne has always been so attentive to the slightest concerns I have had, and she has made my experience with Wiley extremely positive.

Finally, I am of course forever indebted to my mother and my late father.

Problem 1

Cardano and Games of Chance (1564)

Problem. How many throws of a fair die do we need in order to have an even chance of at least one six?

Solution. Let A be the event “a six shows in one throw of a die” and its probability. Then . The probability that a six does not show in one throw is . Let the number of throws be n. Therefore, assuming independence between the throws,

We now solve obtaining , so the number of throws is 4.

1.1 Discussion

In the history of probability, the physician and mathematician Gerolamo Cardano (1501–1575) (Fig. 1.1) was among the first to attempt a systematic study of the calculus of probabilities. Like those of his contemporaries, Cardano's studies were primarily driven by games of chance. Concerning his gambling for 25 years, he famously said in his autobiography (Cardano, 1935, p. 146)

. . .and I do not mean to say only from time to time during those years, but I am ashamed to say it, everyday.

Figure 1.1 Gerolamo Cardano (1501–1575).

Cardano's works on probability were published posthumously in 1663, in the famous 15-page Liber de ludo aleae1 (Fig. 1.2) consisting of 32 small chapters (Cardano, 1564).

Figure 1.2 First page of the Liber de ludo aleae, taken from the Opera Omnia (Vol. I) (Cardano, 1564).

Cardano was undoubtedly a great mathematician of his time but stumbled on the question in Problem 1, and several others too. In this case, he thought the number of throws should be three. In Chapter 9 of his book, Cardano states regarding a die:

One-half of the total number of faces always represents equality2; thus the chances are equal that a given point will turn up in three throws. . .

Cardano's mistake stems from a prevalent general confusion between the concepts of probability and expectation. Let's now dig deeper into Cardano's reasoning. In the Liber, Cardona frequently makes use of an erroneous principle, which Ore calls a “reasoning on the mean” (ROTM) (Ore, 1953, p. 150),3 to deal with various probability problems. According to the ROTM, if an event has a probability p in one trial of an experiment, then in n independent trials the event will occur np times on average, which is then wrongly taken to represent the probability that the event will occur in n trials. For the question in Problem 1, we have p = 1/6 so that, with n = 3 throws, the event “at least a six” is wrongly taken to occur an average np = 3(1/6) = 1/2 of the time (i.e., with probability 1/2).

Using modern notation, let us see why the ROTM is wrong. Suppose an event has a probability p of occurring in a single repetition of an experiment. Then in n independent and identical repetitions of that experiment, the expected number of the times the event occurs is np. Thus, for the die example, the expectation for the number of times a six appears in three throws is 3 × 1/6 = 1/2. However, an expectation of 1/2 in three throws is not the same as a probability of 1/2 in three throws. These facts can formally be seen by using a binomial model.4 Let X be the number of sixes in three throws. Then X has a binomial distribution with parameters n = 3 and p = 1/6, that is, X B(3, 1/6), and its probability mass function is

From this formula, the probability of one six in three throws is

and the probability of at least one six is

Finally, the expected value of X is

which can be simplified to give

Thus, we see that although the expected number of sixes in three throws is 1/2, neither the probability of one six or at least one six is 1/2.

Cardano has not got the recognition that he perhaps deserves for his contributions to the field of probability, for in the Liber de ludo aleae he touched on many rules and problems that were later to become classics. Let us now outline some of these.

In Chapter 14 of the Liber, Cardano gives what some would consider the first definition of classical (or mathematical) probability:

So there is one general rule, namely, that we should consider the whole circuit, and the number of those casts which represents in how many ways the favorable result can occur, and compare that number to the rest of the circuit, and according to that proportion should the mutual wagers be laid so that one may contend on equal terms.

Cardano thus calls the “circuit” what is known as the sample space today, that is, the set of all possible outcomes when an experiment is performed. If the sample space is made up of r outcomes that are favorable to an event, and s outcomes that are unfavorable, and if all outcomes are equally likely, then Cardano correctly defines the odds in favor of the event by . This corresponds to a probability of r/(r + s). Compare Cardano's definition to

The definition given by Leibniz (1646–1716) in 1710 (Leibniz, 1969, p. 161):

If a situation can lead to different advantageous results ruling out each other, the estimation of the expectation will be the sum of the possible advantages for the set of all these results, divided into the total number of results.

Jacob Bernoulli's (1654–1705) statement from the Ars Conjectandi (Bernoulli, 1713, p. 211)5:

. . .if the integral and absolute certainty, which we designate by letter α or by unity 1, will be thought to consist, for example, of five probabilities, as though of five parts, three of which favor the existence or realization of some events, with the other ones, however, being against it, we will say that this event has 3/5α, or 3/5, of certainty.

De Moivre's (1667–1754) definition from the De Mensura Sortis (de Moivre, 1711; Hald, 1984):

If p is the number of chances by which a certain event may happen, and q is the number of chances by which it may fail, the happenings as much as the failings have their degree of probability; but if all the chances by which the event may happen or fail were equally easy, the probability of happening will be to the probability of failing as p to q.

The definition given in 1774 by Laplace (1749–1827), with whom the formal definition of classical probability is usually associated. In his first probability paper, Laplace (1774b) states:

The probability of an event is the ratio of the number of cases favorable to it, to the number of possible cases, when there is nothing to make us believe that one case should occur rather than any other, so that these cases are, for us, equally possible.

However, although the first four definitions (starting from Cardano's) all anteceded Laplace's, it is with the latter that the classical definition was fully appreciated and began to be formally used. In modern notation, if a sample space consists of N equally likely outcomes, of which are favorable to an event A, then Laplace's classical definition of the probability of the event A is

One of Cardano's other important contributions to the theory of probability is “Cardano's formula.”6 Suppose an experiment consists of t equally likely outcomes of which r are favorable to an event. Then the odds in favor of the event in one trial of the experiment are .7 Cardano's formula then states that, in n independent and identical trials of the experiment, the odds in favor of the event occurring n times are .8 While this is an elementary result nowadays, Cardano had some difficulty establishing it.9 At first he thought it was the odds that ought to be multiplied. Cardano calculated the odds against obtaining at least one 1 appearing in a toss of three dice as 125 to 91. Cardano then proceeded to obtain the odds against obtaining at least one 1 in two tosses of three dice as (125/91)2 ≈ 2:1. Thus, on the last paragraph of Chapter 14 of the Liber, Cardano writes (Ore, 1953, p. 202)

Thus, if it is necessary for someone that he should throw an ace twice, then you know that the throws favorable for it are 91 in number, and the remainder is 125; so we multiply each of these numbers by itself and get 8281 and 15,625, and the odds are about 2 to 1.10 Thus, if he should wager double, he will contend under an unfair condition, although in the opinion of some the condition of the one offering double stakes would be better.

However, in the very next chapter entitled On an Error Which Is Made About This, Cardano realizes that it is not the odds that must be multiplied. He comes to understand this by considering an event with odds 1:1 in one trial of an experiment. His multiplication rule for the odds would still give an odds of (1/1)3 = 1:1 for three trials of the experiment, which is clearly wrong. Cardano thus writes (Ore, 1953, pp. 202–203)

But this reasoning seems to be false, even in the case of equality, as, for example, the chance of getting one of any three chosen faces in one cast of one die is equal to the chance of getting one of the other three, but according to this reasoning there would be an even chance of getting a chosen face each time in two casts, and thus in three, and four, which is most absurd. For if a player with two dice can with equal chances throw an even and an odd number, it does not follow that he can with equal fortune throw an even number in each of three successive casts.

Cardano thus correctly calls his initial reasoning “most absurd,” and then gives the following correct reasoning (Ore, 1953):

Therefore, in comparisons where the probability is one-half, as of even faces with odd, we shall multiply the number of casts by itself and subtract one from the product, and the proportion which the remainder bears to unity will be the proportion of the wagers to be staked. Thus, in 2 successive casts we shall multiply 2 by itself, which will be 4; we shall subtract 1; the remainder is 3; therefore a player will rightly wager 3 against 1; for if he is striving for odd and throws even, that is, if after an even he throws either even or odd, he is beaten, or if after an odd, an even. Thus he loses three times and wins once.

Cardano thus realizes that it is the probability, not the odds, that ought to be multiplied.11 However, in the very next sentence following his previous correct reasoning, he makes a mistake again when considering three consecutive casts for an event with odds 1:1. Cardano wrongly states that the odds against the event happening in three casts are 1:(32 − 1) = 1:8, instead of 1:(23 − 1) = 1:7. Nevertheless, further in the book, Cardano does give the correct general rule (Ore, 1953, p. 205):

Thus, in the case of one die, let the ace and the deuce be favorable to us; we shall multiply 6, the number of faces, into itself: the result is 36; and two multiplied into itself will be 4; therefore the odds are 4 to 32, or, when inverted, 8 to 1.

If three throws are necessary, we shall multiply 3 times; thus, 6 multiplied into itself and then again into itself gives 216; and 2 multiplied into itself and again into 2, gives 8; take away 8 from 216: the result will be 208; and so the odds are 208 to 8, or 26 to 1. And if four throws are necessary, the numbers will be found by the same reasoning, as you see in the table; and if one of them be subtracted from the other, the odds are found to be 80 to 1.

In the above, Cardano has considered an event with probability 1/3, and correctly gives the odds against the event happening twice as (32 − 1):1 = 8:1, happening thrice as (33 − 1):1 = 26:1, and so on. Cardano thus finally reaches the following correct rule: if the odds in favor of an event happening in one trial of an experiment are , then in n independent and identical trials of the experiment, the odds against the event happening n times are .

Cardano also anticipated the law of large numbers (see Problem 8), although he never explicitly stated it. Ore writes (1953, p. 170)

It is clear. . .that he [Cardano] is aware of the so-called law of large numbers in its most rudimentary form. Cardano's mathematics belongs to the period antedating the expression by means of formulas, so that he is not able to express the law explicitly in this way, but he uses it as follows: when the probability for an event is p then by a large number n of repetitions the number of times it will occur does not lie far from the value m = np.

Moreover, in Chapter 11 of the De ludo aleae, Cardano investigated the Problem of Dice (see Problem 3); in the Practica arithmetice (Cardano, 1539), he discussed the Problem of Points (see Problem 4), the Gambler's Ruin (see Problem 5) (Coumet, 1965a), and the St Petersburg Problem (see Problem 11) (Dutka, 1988); finally, in the Opus novum de proportionibus (Cardano, 1570), Cardano also made use of what later came to be known as Pascal's Arithmetic Triangle (see Problem 4) (Boyer, 1950). However, in none of these problems did Cardano reach the level of mathematical sophistication and maturity that was later to be evidenced in the hands of his successors.

We make a final comment on Cardano's investigations in probability. From the Liber de ludo aleae it is clear that Cardano is unable to disassociate the unscientific concept of luck from the mathematical concept of chance. He identifies luck with some supernatural force that he calls the “authority of the Prince” (Ore, 1953, p. 227). In Chapter 20 entitled On Luck in Play, Cardano states (Ore, 1953, pp. 215–216)

In these matters, luck seems to play a very great role, so that some meet with unexpected success while others fail in what they might expect. . .

If anyone should throw with an outcome tending more in one direction than it should and less in another, or else it is always just equal to what it should be, then, in the case of a fair game there will be a reason and a basis for it, and it is not the play of chance; but if there are diverse results at every placing of the wagers, then some other factor is present to a greater or less extent; there is no rational knowledge of luck to be found in this, though it is necessarily luck.

Cardano thus believes that there is some external force that is responsible for the fluctuations of outcomes from their expectations. He fails to recognize such fluctuations are germane to chance and not because of the workings of supernatural forces. Gigerenzer et al. (1989, p. 12) thus write

. . .He [Cardano] thus relinquished his claim to founding the mathematical theory of probability. Classical probability arrived when luck was banished; it required a climate of determinism so thorough as to embrace even variable events as expressions of stable underlying probabilities, at least in the long run.

Notes

1. The Book on Games of Chance. An English translation of the book and a thorough analysis of Cardano's connections with games of chance can be found in Ore's Cardano: The Gambling Scholar (Ore, 1953). More bibliographic details can be found in Gliozzi (1980, pp. 64–67) and Scardovi (2004, pp. 754–758).

2. Cardano frequently uses the term “equality” in the Liber to denote half of the total number of sample points in the sample space. See Ore (1953, p. 149).

3. See also Williams (2005).

4. For the origin of the binomial model, see Problem 8.

5. The translation that follows is taken from Oscar Sheynin's translations of Chapter 4 of the Ars Conjectandi (Sheynin, 2005).

6. Not to be confused with the other Cardano's formula having to do with the general solution of a “reduced” cubic equation (i.e., a cubic equation with no second-degree term). Cardano also provided methods to convert the general cubic equation to the reduced one. These results appeared in Cardano's opus Ars Magna (The Great Art) and had been communicated to him previously by the mathematician Niccolò Tartaglia of Brescia (1499–1557) after swearing that he would never disclose the results. A bitter dispute thereby ensued between Cardano and Tartaglia, and is nicely documented in Hellman's Great Feuds in Mathematics (Hellman, 2006, pp. 7–25).

7. Thus, the odds against the event in one trial are (t − r): r.

8. This is the same as saying that, if an event has probability p (=r/t) of occurring in one trial of an experiment, then the probability that it will occur in all of n independent and identical trials of the experiment is pn.

9. See also Katz (1998, p. 450).

10. The odds calculated by Cardano are the odds against the event in question.

11. Thus for the 125:91 example, the correct odds against in two trials are (2162 − 912)/912 ≈ 4.63:1 and for the 1:1 example, the correct odds in favor in three trials are 13:(23 − 13) = 1:7.

Problem 2

Galileo and a Discovery Concerning Dice (1620)

Problem. Suppose three dice are thrown and the three numbers obtained added. The total scores of 9, 10, 11, and 12 can all be obtained in six different combinations. Why then is a total score of 10 or 11 more likely than a total score of 9 or 12?

Solution. Table 2.1 shows each of the six possible combinations (unordered arrangements) for the scores of 9–12. Also shown is the number of ways (permutations or ordered arrangements) in which each combination can occur.

Table 2.1 Combinations and Number of Ways Scores of 9–12 that Can Be Obtained When Three Dice are Thrown.

For example, reading the first entry under the column 12, we have a 6-5-1. This means that, to get a total score of 12, one could get a 6, 5, 1 in any order. Next to the 6-5-1 is the number 6. This is the number of different orders in which one can obtain a 6, 5, 1. Hence, we see that the scores of 9–12 can all be obtained using six combinations for each. However, because different combinations can be realized in a different number of ways, the total number of ways for the scores 9, 10, 11, and 12 are 25, 27, 27, and 25, respectively. Hence, scores of 10 or 11 are more likely than scores of 9 or 12.

2.1 Discussion

Almost a century after Cardano's times, this problem was asked by the Grand Duke of Tuscany to the renowned physicist and mathematician Galileo Galilei (1564–1642) (Fig. 2.1). The throwing of three dice was part of the game of passadieci, which involved adding up the three numbers and getting at least 11 points to win. Galileo gave the solution in his probability paper Sopra le scoperte dei dadi1 (Galilei, 1620) (see Fig. 2.2). In his paper, Galileo states (David, 1962, p. 193)

But because the numbers in the combinations in three-dice throws are only 16, that is, 3, 4, 5, etc. up to 18, among which one must divide the said 216 throws, it is necessary that to some of these numbers many throws must belong; and if we can find how many belong to each, we shall have prepared the way to find out what we want to know, and it will be enough to make such an investigation from 3 to 10, because what pertains to one of these numbers, will also pertain to that which is the one immediately greater.

Figure 2.1 Galileo Galilei (1564–1642).

Figure 2.2 Extract from Galileo's article “Sopra le scoperte dei dadi,” taken from Le Opere di Galileo XIV (Galilei, 1620). The Sopra here appears under the name “Considerazione sopra il giuoco dei dadi.”

Galileo then proceeds to use a method similar to the one in the solution provided previously. The almost casual way in which he counts the number of favorable cases from the total number of equally possible outcomes indicates that the use of the classical definition of probability was common at that time. Unbeknownst to Galileo, the same problem had actually already been successfully solved by Cardano almost a century earlier. The problem appeared in Chapter 13 of Cardano's Liber de ludo aleae (Ore, 1953, p. 198), which was published 21 years after Galileo's death (see Fig. 2.3).

Figure 2.3 Cardano's solution of the problem considered by Galileo, as it appears in Chapter 13 of the Liber de ludo aleae (Cardano, 1564). The bottom left column on the right page has the two last rows reading 9, 12, 25 and 10, 11, 27. These correspond, respectively, to a total of 25 ways of obtaining a total of 9 or 12 with three dice, and a total of 27 ways of obtaining a total of 10 or 11 with three dice.

Note

1. On a Discovery Concerning Dice.

Problem 3

The Chevalier de Méré Problem I: The Problem of Dice (1654)

Problem. When a die is thrown four times, the probability of obtaining at least one six is a little more than 1/2. However, when two dice are thrown 24 times, the probability of getting at least one double-six is a little less than 1/2. Why are the two probabilities not the same, given the fact that Pr{double-six for a pair of dice} = 1/36 =1/6·Pr{a six for a single die}, and you compensate for the factor of 1/6 by throwing 6 · 4 = 24 times when using two dice?

Solution. Both probabilities can be calculated by using the multiplication rule of probability. In the first case, the probability of no six in one throw is 1 − 1/6 = 5/6. Therefore, assuming independence between the throws,

In the second case, the probability of no double-six in one throw of two dice is 1 − (1/6)2 = 35/36. Therefore, again assuming independence,

3.1 Discussion

It is fairly common knowledge that the gambler Antoine Gombaud (1607–1684), better known as the Chevalier de Méré,1 had been winning consistently by betting even money that a six would come up at least once in four rolls with a single die. However, he had now been losing other bets, when in 1654 he met his friend, the amateur mathematician Pierre de Carcavi (1600–1684). This was almost a quarter century after Galileo's death. De Méré had thought the odds were favorable on betting that he could throw at least one sonnez (i.e., double-six) with 24 throws of a pair of dice. However, his own experiences indicated that 25 throws were required.2 Unable to resolve the issue, the two men consulted their mutual friend, the great mathematician, physicist, and philosopher Blaise Pascal (1623–1662).3 Pascal himself had previously been interested in the games of chance (Groothuis, 2003, p. 10). Pascal must have been intrigued by this problem and, through the intermediary of Carcavi,4 contacted the eminent mathematician, Pierre de Fermat (1601–1665),5 who was a lawyer in Toulouse. Pascal knew Fermat through the latter's friendship with Pascal's father, who had died 3 years earlier. The ensuing correspondence, albeit short, between Pascal and Fermat is widely believed to be the starting point of the systematic development of the theory of probability. In the first extant letter6 Pascal addressed to Fermat, dated July 29, 1654, Pascal says (Figs. 3.1 and 3.2) (Smith, 1929, p. 552)

He [De Méré] tells me that he has found an error in the numbers for this reason:

If one undertakes to throw a six with a die, the advantage of undertaking to do it in 4 is as 671 is to 625.

If one undertakes to throw double sixes with two dice the disadvantage of the undertaking is 24.

But nonetheless, 24 is to 36 (which is the number of faces of two dice) as 4 is to 6 (which is the number of faces of one die).

This is what was his great scandal which made him say haughtily that the theorems were not consistent and that arithmetic was demented. But you can easily see the reason by the principles which you have.

Figure 3.1 Extract from first extant letter from Pascal to Fermat, taken from Smith's A Source Book in Mathematics (Smith, 1929).

Figure 3.2 Extract from first extant letter from Fermat to Pascal, taken from Smith's A Source Book in Mathematics (Smith, 1929).

De Méré was thus distressed that his observations were in contradiction with his mathematical calculations. To Fermat, however, the Problem of Dice was an elementary exercise that he solved without trouble, for Pascal says in his July 29 letter (Smith, 1929, p. 547)

I do not have the leisure to write at length, but, in a word, you have found the two divisions. . .of the dice with perfect justice. I am thoroughly satisfied as I can no longer doubt that I was wrong, seeing the admirable accord in which I find myself with you.

On the other hand, de Méré's erroneous mathematical reasoning was based on the incorrect Old Gambler's Rule (Weaver, 1982, p. 47), which uses the concept of the critical value of a game. The critical value C of a game is the smallest number of plays such that the probability the gambler will win at least one play is 1/2 or more. Let us now explain how the Old Gambler's Rule is derived. Recall Cardano's “reasoning on the mean” (ROTM, see Problem 1): if a gambler has a probability p of winning one play of a game, then in n independent plays the gambler will win an average of np times, which is then wrongly equated to the probability of winning in n plays. Then, by setting the latter probability to be half, we have

By substituting p = 1/36 in the above formula, Cardano had obtained a wrong answer of C = 18 for the number of throws. Furthermore, given a first game with, then a second game which has probability of winning in each play must have critical value , where

(3.1)

That is, the Old Gambler's Rule states that the critical values of two games are in inverse proportion to their respective probabilities of winning. Using , we get . But we have seen that, with 24 throws, the probability of at least one double-six is .491, which is less than 1/2. So cannot be a critical value (the correct critical value is shown later to be 25), and the Old Gambler's Rule cannot be correct. It was thus the belief in the validity of the Old Gambler's Rule that made de Méré wrongly think that, with 24 throws, he should have had a probability of 1/2 for at least one double-six.

Digging further, let us see how the erroneous Old Gambler's Rule should be corrected. By definition, , the smallest integer greater than or equal to , such that , that is, . With obvious notation, for the second game: , where . Thus, the true relationship should be

(3.2)

We see that Eqs. (3.1) and (3.2) are quite different from each other. Even if p1 and p2 were very small, so that and , we would get approximately. This is still different from Eq. (3.1) because the latter uses the integers and , instead of the real numbers and .

The Old Gambler's Rule was later investigated by the renowned French mathematician Abraham de Moivre (1667–1754), who was a close friend to Isaac Newton. In his Doctrine of Chances (de Moivre, 1718, p. 14), de Moivre solves and obtains . For small p,

(3.3)

(see Fig. 3.3). Let us see if we obtain the correct answer when we apply de Moivre's Gambling Rule for the two-dice problem. Using with p = 1/36 gives x ≈ 24.95 and we obtain the correct critical value C = 25. The formula works only because p is small enough and is valid only for such cases.7 The other formula that could be used, and that is valid for all values of p, is . For the two-dice problem, this exact formula gives = 24.60 so that = 25. Table 3.1 compares critical values obtained using the Old Gambler's Rule, de Moivre's Gambling Rule, and the exact formula.

Figure 3.3 Extract from de Moivre's derivation of the gambling rule, taken from the first edition of the Doctrine of Chances (de Moivre, 1718, p. 14). Note that de Moivre defines q to be b/a or (1/p) − 1, whereas in the text we use q = 1 − p.

Table 3.1 Critical Values Obtained Using the Old Gambling Rule, de Moivre's Gambling Rule and the Exact Formula for Different Values of p, the Probability of the Event of Interest.

We next use de Moivre's Gambling Rule to solve the following classic problem:

A gambler has a probability of 1/N of winning a game, where N is large. Show that she must play the game about (2/3)N times in order to have a probability of at least 1/2 of winning at least once.

To solve this problem, note that p = 1/N is small so that we can apply de Moivre's Gambling Rule in Eq. (3.3):

as required.

As a final note on the dice problem, Pascal himself never provided a solution to it in his known communication with Fermat. He had undoubtedly thought the problem was very simple. However, he devoted much of his time and energy on the next classic problem.

Notes

1. Leibniz describes the Chevalier de Méré as “a man of penetrating mind who was both a player and a philosopher” (Leibniz, 1896, p. 539). Pascal biographer Tulloch also notes (1878, p. 66): “Among the men whom Pascal evidently met at the hotel of the Duc de Roannez [Pascal's younger friend], and with whom he formed something of a friendship, was the well-known Chevalier de Méré, whom we know best as a tutor of Madame de Maintenon, and whose graceful but flippant letters still survive as a picture of the time. He was a gambler and libertine, yet with some tincture of science and professed interest in its progress.” Pascal himself was less flattering. In a letter to Fermat, Pascal says (Smith, 1929, p. 552): “. . .he [de Méré] has ability but he is not a geometer (which is, as you know, a great defect) and he does not even comprehend that a mathematical line is infinitely divisible and he is firmly convinced that it is composed of a finite number of points. I have never been able to get him out of it. If you could do so, it would make him perfect.” The book by Chamaillard (1921) is completely devoted to the Chevalier de Méré.

2. Ore (1960) believes that the difference in the probabilities for 24 and 25 throws is so small that it is unlikely that de Méré could have detected this difference through observations. On the other hand, Olofsson (2007, p. 177) disagrees. With 24 throws, the casino would consistently make a profit of 2% (51–49%) and other gamblers would also pay for it. Furthermore, he points out that, if de Méré started with 100 pistoles (gold coins used in Europe in the seventeenth and eighteenth centuries), he would go broke with probability .97 before doubling, if the true probability is .49. If the true probability is .51, de Méré would double before going broke with very high probability. Thus, Olofsson contends it is possible to detect a difference between .49 and .51 through actual observations, and de Méré had enough time at his disposal for this.

3. Of the several books that have been written on Pascal, the biographies by Groothuis (2003) and Hammond (2003) are good introductions to his life and works.

4. Carcavi had been an old friend of Pascal's father and was very close to Pascal.

5. Fermat is today mostly remembered for the so-called “Fermat Last Theorem”, which he conjectured in 1637 and which was not proved until 1995 by Andrew Wiles (1995). The theorem states that no three positive integers a, b, c can satisfy the equation an + bn = cn for any integer n greater than 2. A good introduction to Fermat's Last Theorem can be found in Aczel (1996). The book by Mahoney (1994) is an excellent biography of Fermat, whose probability work appears on pp. 402–410 of the book.

6. Unfortunately, the very first letter Pascal wrote to Fermat, as well as a few other letters between the two men, no longer exists. However, see the most unusual book by Rényi (1972), in which he fictitiously reconstructs four letters between Pascal and Fermat.

7. For example, if we apply de Moivre's Gambling Rule to the one-die problem, we obtain x = .693/(1/6) = 4.158 so that C = 5. This cannot be correct because we showed in the solution that we need only 4 tosses.

Problem 4

The Chevalier de Méré Problem II: The Problem of Points (1654)

ProblemTwo players A and B play a fair game such that the player who wins a total of 6 rounds first wins a prize. Suppose the game unexpectedly stops when A has won a total of 5 rounds and B has won a total of 3 rounds. How should the prize be divided between A and B?

Solution. The division of the prize is determined by the relative probabilities of A and B winning the prize, had they continued the game. Player A is one round short, and player B three rounds short, of winning the prize. The maximum number of hypothetical remaining rounds is (1 + 3) − 1 = 3, each of which could be equally won by A or B. The sample space for the game is Ω = {A1, B1A2, B1B2A3, B1B2B3}. Here B1A2, for example, denotes the event that B would win the first remaining round and A would win the second (and then the game would have to stop since A is only one round short). However, the four sample points in Ω are not equally likely. Event A1 occurs if any one of the following four equally likely events occurs: A1A2A3, A1A2B3, A1B2A3, and A1B2B3. Event B1A2 occurs if any one of the following two equally likely events occurs: B1A2A3 and B1A2B3. In terms of equally likely sample points, the sample space is thus

There are in all eight equally likely outcomes, only one of which (B1B2B3) results in B hypothetically winning the game. Player A thus has a probability 7/8 of winning. The prize should therefore be divided between A and B in the ratio 7:1.

4.1 Discussion

We note that the sample space for this game is not based on how many rounds each player has already won. Rather it depends on the maximum number of remaining rounds that could be played.

Problem 4, also known as the Problem of Points1 or the Division of Stakes Problem, was another problem de Méré asked Pascal (Fig. 4.1) in 1654. The problem had already been known hundreds of years before the times of these mathematicians.2 It had appeared in Italian manuscripts as early as 1380 (Burton, 2006, p. 445). However, it first came in print in Fra Luca Pacioli's (1494) Summa de arithmetica, geometrica, proportioni, et proportionalita.3 Pacioli's incorrect answer was that the prize should be divided in the same ratio as the total number of rounds the players had won. Thus, for Problem 4, the ratio is 5:3. A simple counterexample shows why Pacioli's reasoning cannot be correct. Suppose players A and B need to win 100 rounds to win a game, and when they stop A has won one round and B has won none. Then Pacioli's rule would give the whole prize to A even though he is a single round ahead of B and would have needed to win 99 more rounds had the game continued!4

Figure 4.1 Blaise Pascal (1623–1662).

Cardano had also considered the Problem of Points in the Practica arithmetice (Cardano, 1539). His major insight was that the division of stakes should depend on how many rounds each player had yet to win, not on how many rounds they had already won. However, in spite of this, Cardano was unable to give the correct division ratio: he concluded that, if players A and B are a and b rounds short of winning, respectively, then the division ratio between A and B should be b(b + 1):a(a + 1). In our case, a = 1, b = 3, giving a division ratio of 6:1.

At the opening of his book Recherches sur la Probabilité des Jugements en Matières Criminelles et Matière Civile, the distinguished mathematician Siméon Denis Poisson (1781–1840) pronounced these famous words (Poisson, 1837, p. 1):

A problem relating to the games of chance, proposed to an austere Jansenist [Pascal] by a man of the world [de Méré], was at the origin of the calculus of probabilities. Its objective was to determine the proportion into which the stake should be divided between the players, when they decided to stop playing. . .

Poisson's words echo the still widely held view today that probability essentially sprung from considerations of games of chance during the Enlightenment. We should mention, however, that several other authors have a different viewpoint. For example, Maistrov (1974, p. 7) asserts:

Up to the present time there has been a widespread false premise that probability theory owes its birth and early development to gambling.

Maistrov explains that gambling existed since ancient and medieval times, but probability did not develop then. He contends that its development occurred in the sixteenth and seventeenth century owing to economic developments, resulting in an increase in monetary transactions and trade. This seems to indeed be the case. In a recent work, Courtebras is of a similar opinion when he says (Courtebras, 2008, p. 51)

. . .behind the problems in games of chance which enable the elaboration of easily quantifiable solutions, new attitudes are being formed toward a world characterized by the development of towns and money, of productions and exchanges.

It is also important to note that the initial works in the field, from the times of Cardano until those before Jacob Bernoulli, were primarily concerned with problems of fairness or equity. Historian Ernest Coumet has thus noted that the Problem of Points has a judicial origin (Coumet, 1970). With the Renaissance came the legalization of both games of chance and aleatory contracts. By law, the latter had to be fair to either party. The Problem of Points is thus a model for the repartition of gains in arbitrary situations of uncertainty, mainly characterized by the notion of equity. As Gregersen (2011, p. 25) explains

. . .The new theory of chances was not, in fact, simply about gambling but also about the legal notion of a fair contract. A fair contract implied equality of expectations, which served as the fundamental notion in these calculations. Measures of chance or probability were derived secondarily from these expectations.

Probability was tied up with questions of law and exchange in one other crucial respect. Chance and risk, in aleatory contracts, provided a justification for lending at interest, and hence a way of avoiding Christian prohibitions against usury. Lenders, the argument went, were like investors; having shared the risk, they deserved also to share in the gain. For this reason, ideas of chance had already been incorporated in a loose, largely nonmathematical way into theories of banking and marine insurance.

An additional interesting point is that, throughout their correspondence, although both Pascal and Fermat (Fig. 4.2) were deeply engaged in calculating probabilities, they never actually used the word “probability” in their investigations. Instead they talked about division ratios and used such terms as “value of the stake” or “value of a throw” to express a player's probability of winning. The term “probability” as a numerical measure was actually first used in Antoine Arnauld's (1612–1694) and Pierre Nicole's (1625–1695) widely influential La Logique, ou l'Art de Penser5 (Arnauld and Nicole, 1662).

Figure 4.2 Pierre de Fermat (1601–1665).

As a final observation, before we dig deeper into Pascal's and Fermat's solutions to the Problem of Points, we comment on the significance of their communication. It is generally believed that the theory of probability really started through the correspondence of these mathematicians, although this notion has occasionally been disputed by some. For example, in his classic work Cardano: the Gambling Scholar (1953), Ore made a detailed study of Cardano's Liber de ludo aleae and pointed out (p. viii)

. . .I have gained the conviction that this pioneer work on probability is so extensive and in certain questions so successful that it would seem much more just to date the beginnings of probability theory from Cardano's treatise rather than the customary reckoning from Pascal's discussions with his gambling friend de Méré and the ensuing correspondence with Fermat, all of which took place at least a century after Cardano began composing his De ludo aleae.

Burton seems to share the same opinion, for he says (Burton, 2006, p. 445)

For the first time, we find a transition from empiricism to the theoretical concept of a fair die. In making it, Cardan [Cardano] probably became the real father of modern probability theory.

On the other hand, Edwards (1982) is quite categorical:

. . .in spite of our increased awareness of the earlier work of Cardano (Ore, 1953) and Galileo (David, 1962) it is clear that before Pascal and Fermat no more had been achieved than the enumeration of the fundamental probability set in various games with dice or cards.

Although Cardano was the first to study probability, it was Pascal's and Fermat's work however that provided the first impetus for a systematic study and development of the mathematical theory of probability. Likewise, contemporaries of Cardano such as Pacioli, Tartaglia, and Peverone did consider probability calculations involving various games of chance. However, as Gouraud (1848, p. 3) explains

. . .but these crude essays, consisting of extremely erroneous analyses and having all remained equally sterile, do not merit the consideration of either critiques or history. . .

Let us now discuss Pascal's and Fermat's individual contributions to the Problem of Points. Pascal was at first unsure of his own solution to the problem, and turned to a friend, the mathematician Gilles Personne de Roberval (1602–1675). Roberval was not of much help, and Pascal then asked for the opinion of Fermat, who was immediately intrigued by the problem. A beautiful account of the ensuing correspondence between Pascal and Fermat can be found in a recent book by Keith Devlin, The Unfinished Game: Pascal, Fermat and the Seventeenth Century Letter That Made the World Modern (Devlin, 2008). An English translation of the extant letters can be found in Smith (1929, pp. 546–565). In a letter dated August 24, 1654, Pascal says (Smith, 1929, p. 554)

I wish to lay my whole reasoning before you, and to have you do me the favor to set me straight if I am in error or to endorse me if I am correct. I ask you this in all faith and sincerity for I am not certain even that you will be on my side.

Fermat made use of the fact that the solution depended not on how many rounds each player had already won but on how many each player must still win to win the prize. This is the same observation Cardano had previously made, although he had been unable to solve the problem correctly.

The solution we provided earlier is based on Fermat's idea of extending the unfinished game. Fermat also enumerated the different sample points like in our solution and reached the correct division ratio of 7:1.

Pascal seems to have been aware of Fermat's method of enumeration (Edwards, 1982), at least for two players, and also believed there was a better method. For he says in his first extant letter, dated July 29, 1654 (Smith, 1929, p. 548)