Classic Problems of Probability

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.

Preface ix

Acknowledgments xi

1 Cardano and Games of Chance (1564) 1

2 Galileo and a Discovery Concerning Dice (1620) 9

3 The Chevalier de Méré Problem I: The Problem of Dice (1654) 13

4 The Chevalier de Méré Problem II: The Problem of Points (1654) 20

5 Huygens and the Gambler’s Ruin (1657) 39

6 The Pepys–Newton Connection (1693) 49

7 Rencontres with Montmort (1708) 54

8 Jacob Bernoulli and his Golden Theorem (1713) 62

9 De Moivre’s Problem (1730) 81

10 De Moivre, Gauss, and the Normal Curve (1730, 1809) 89

11 Daniel Bernoulli and the St. Petersburg Problem (1738) 108

12 d’Alembert and the “Croix ou Pile” Article (1754) 119

13 d’Alembert and the Gambler’s Fallacy (1761) 124

14 Bayes, Laplace, and Philosophies of Probability (1764, 1774) 129

15 Leibniz’s Error (1768) 156

16 The Buffon Needle Problem (1777) 159

17 Bertrand’s Ballot Problem (1887) 169

18 Bertrand’s Strange Three Boxes (1889) 175

19 Bertrand’s Chords (1889) 179

20 Three Coins and a Puzzle from Galton (1894) 186

21 Lewis Carroll’s Pillow Problem No. 72 (1894) 189

22 Borel and a Different Kind of Normality (1909) 194

23 Borel’s Paradox and Kolmogorov’s Axioms (1909, 1933) 199

24 Of Borel, Monkeys, and the New Creationism (1913) 208

25 Kraitchik’s Neckties and Newcomb’s Problem (1930, 1960) 215

26 Fisher and the Lady Tasting Tea (1935) 224

27 Benford and the Peculiar Behavior of the First Significant Digit (1938) 233

28 Coinciding Birthdays (1939) 240

29 Lévy and the Arc Sine Law (1939) 247

30 Simpson’s Paradox (1951) 253

31 Gamow, Stern, and Elevators (1958) 260

32 Monty-Hall, Cars, and Goats (1975) 264

33 Parrondo’s Perplexing Paradox (1996) 271

Bibliography 277

Photo Credits 296

Index 299

PRAKASH GORROOCHURN, PhD, is Assistant Professor in the Department of Biostatistics at Columbia University, where he is also a statistical consultant in the School of Social Work. Dr. Gorroochurn has published extensively in his areas of research interest, which include mathematical population genetics and genetic epidemiology.