The Mathematics of Games and Gambling

It's often said that you have the same chance of winning the lottery, whether or not you buy a ticket. While that statement is not absolutely true, the infinitesimal probability of winning is close enough to zero to dissuade anyone who understands the concept of expectation from playing the state-run lotteries. However, the general public has not yet caught on: in weeks when the jackpot offered is unusually high, so many New Yorkers travel to Connecticut to buy Powerball tickets that Metro-North (a local commuter rail line) distributes fliers indicating at which train stops the tickets (not available in New York State) can be purchased.

If some of those eager Powerball players invested their money in Edward Packel's The Mathematics of Games and Gambling, the money they would save by forgoing such loser's bets would soon cover the purchase price of the book. Not that I have anything against gambling per se, but some of the earlier research into statistics was conducted in support of gambling, and the current popular interest in poker and other games of chance provides an excellent teaching opportunity to introduce elementary concepts of probability.

Consider the case of the Chevalier de Méré, who thought that because he prospered by making bets at even odds that he would roll at least one six in four rolls of the dice, he should do equally well betting that he would roll two or more double sixes in twenty-four rolls of two dice. In fact, the first bet is a consistent winner and the second is a consistent loser, and in their efforts to explain why this was so, the mathematicians Blaise Pascal, and Pierre de Fermat developed the concepts of binomial probability and what is now known as "Pascal's Triangle". To his credit, Packel notes although Pascal and Fermat brought this knowledge to the Western world, the triangle was known earlier in both China and Persia, where it is known as the "Khayyam triangle" after the Persian poet and astronomer Omar Khayyam. Inclusion of such information is a significant strength of Packel's book: the mathematical content is embedded in a social and cultural context which makes it enjoyable reading even for individuals who have no vested interest in either mathematics or gambling. The first chapter, which combines a discussion of Girolamo Cardano and Fyodor Dostoyesvsky, should win over the most right-brained English major.

The Mathematics of Games and Gambling is suitable for many different audiences. Those who have a recreational interest in mathematics will enjoy reading it because of the clear explanations of the mathematics behind many popular games of chance, and those explanations require only a knowledge of high school mathematics. Beginning students of probability will appreciate this book for the same reason: Packel's exposition of basic probability is clearer than that contained in some textbooks. It would be an ideal textbook or supplement in courses for mathematics for non-majors. And those who are truly interested in gambling should buy a copy in order to understand how to optimize their winnings, or at least lose their money more slowly. In fact, I'm beginning to think of Packel as an ambassador to the non-mathematical majority, demonstrating to them that mathematics can be useful and entertaining, and that mathematicians can be literate and articulate.

Edward W. Packel is currently the Volwiler Professor of Mathematics at Lake Forest College in suburban Chicago, where he has taught mathematics and computer science courses since 1971. Packel received his BA from Amherst College in 1963 and his PhD from MIT in 1967. His research interests include game theory and social choice theory, functional analysis, information-based complexity, and the use of technology such as Mathematica in teaching mathematics. Professor Packel maintains a website at http://math.lfc.edu/packel/.

Sarah Boslaugh, PhD, MPH, is a Performance Analyst for BJC HealthCare in Saint Louis, Missouri. She published An Intermediate Guide to SPSS Programming with Sage in 2005 and is currently editing The Encyclopedia of Epidemiology for Sage (forthcoming, 2007) and writing Secondary Data Sources for Public Health (forthcoming, 2007) for Cambridge University Press.