Classic Problems of Probability

"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 Cl assic 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 3 Problem 1. Cardano and Games of Chance (1564) 8 Problem 2. Gailieo and a Discovery Concerning Dice (1620) 15 Problem 3. The Chevalier de Méré Problem I: The Problem of Dice (1654) 17 Problem 4. The Chevalier de   Méré Problem II: The Problem of Points (1654) 22 Problem 5. Huygens and the Gambler’s Ruin (1657) 39 Problem 6. The Pepys–Newton Connection (1693) 47 Problem 7. Rencontres with Montmort (1708) 50 Problem 8. Jacob Bernoulli and his Golden Theorem (1713) 54 Problem 9. De Moivre’s Problem (1730) 71 Problem 10. De Moivre, Gauss, and the Normal Curve (1730, 1809) 79 Problem 11. Daniel Bernoulli and the St Petersburg Problem (1738) 94 Problem 12. D’Alembert and the “Croix ou Pile” Article (1754) 102 Problem 13. D’Alembert and the Gambler’s Fallacy (1761) 105 Problem 14. Bayes, Laplace, and Philosophies of Probability (1764, 1774) 109 Problem 15. Leibniz’s Error (1768) 132 Problem 16. The Buffon Needle Problem (1777) 134 Problem 17. Bertrand’s Ballot Problem (1887) 143 Problem 18. Bertrand’s Strange Three Boxes (1889) 147 Problem 19. Bertrand’s Chords (1889) 151 Problem 20. Three Coins and a Puzzle from Galton (1894) 156 Problem 21. Lewis Carroll’s Pillow Problem No. 72 (1894) 157 Problem 22. Borel and A Different Kind of Normality (1909) 161 Problem 23. Borel’s Paradox and Kolmogorov’s Axioms (1909, 1933) 165 Problem 24. Of Borel, Monkeys, and the New Creationism (1913) 173 Problem 25. Kraitchik’s Neckties and Newcomb’s Problem (1930, 1960) Problem 26. Fisher and the lady Tasting Tea (1935) 188 Problem 27. Benford and the Peculiar Behavior of the First Significant Digit (1938) 195 Problem 28. Coinciding Birthdays (1939) 200 Problem 29. Lévy and the Arc Sine Law (1939) 206 Problem 30. Simpson’s Paradox (1951) 210 Problem 31. Gamow, Stern, and Elevators (1958) 215 Problem 32. Monty–Hall, Cars, and Goats (1975) 218 Problem 33. Parrondo’s Perplexing Paradox (1996) 224 Bibliography 230 Photo Credits 254

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.

“Thus, the book can be highly recommend to every lecturer in this field and every student interested in probability and statistics.”  ( Zentralblatt Math , 1 September   2013)