Classic Problems of Probability

Prakash Gorroochurn

• Publisher: Wiley
• Year: 2012
• ISBN:
  9781118314333
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

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

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.

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 p...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• Preface
• Acknowledgments
• Problem 1: Cardano and Games of Chance (1564)
• Problem 2: Galileo and A Discovery Concerning Dice (1620)
• Problem 3: The Chevalier de Méré Problem I: The Problem of Dice (1654)
• Problem 4: The Chevalier de Méré Problem II: The Problem of Points (1654)
• Problem 5: Huygens and the Gambler's Ruin (1657)
• Problem 6: The Pepys–Newton Connection (1693)
• Problem 7: Rencontres with Montmort (1708)
• Problem 8: Jacob Bernoulli and His Golden Theorem (1713)
• Problem 9: De Moivre's Problem (1730)
• Problem 10: De Moivre, Gauss, and the Normal Curve (1730, 1809)
• Problem 11: Daniel Bernoulli and the St: Petersburg Problem (1738)
• Problem 12: D'Alembert and the “Croix ou Pile” Article (1754)
• Problem 13: D'Alembert and the Gambler's Fallacy (1761)
• Problem 14: Bayes, Laplace, and Philosophies of Probability (1764, 1774)
• Problem 15: Leibniz's Error (1768)
• Problem 16: The Buffon Needle Problem (1777)
• Problem 17: Bertrand's Ballot Problem (1887)
• Problem 18: Bertrand's Strange Three Boxes (1889)
• Problem 19: Bertrand's Chords (1889)
• Problem 20: Three Coins and A Puzzle from Galton (1894)
• Problem 21: Lewis Carroll's Pillow Problem No: 72 (1894)
• Problem 22: Borel and A Different Kind of Normality (1909)
• Problem 23: Borel's Paradox and Kolmogorov's Axioms (1909, 1933)
• Problem 24: Of Borel, Monkeys, and the New Creationism (1913)
• Problem 25: Kraitchik's Neckties and Newcomb's Problem (1930, 1960)
• Problem 26: Fisher and the Lady Tasting Tea (1935)
• Problem 27: Benford and the Peculiar Behavior of the First Significant Digit (1938)
• Problem 28: Coinciding Birthdays (1939)
• Problem 29: Lévy and the Arc Sine Law (1939)
• Problem 30: Simpson's Paradox (1951)
• Problem 31: Gamow, Stern, and Elevators (1958)
• Problem 32: Monty Hall, Cars, and Goats (1975)
• Problem 33: Parrondo's Perplexing Paradox (1996)
• Bibliography
• Photo Credits
• Index