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