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Chances Are... Adventures in Probability

Michael Kaplan and Ellen Kaplan
Publication Date: 
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[Reviewed by
Andrew M. Ross
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Chances Are…  discusses many social issues where probability and statistics play (or should play) key roles. Its approach is much more literary, philosophical, and historical than most other books in this genre. It starts by introducing many of the usual probability basics, such as dice games, Pascal's triangle, Bayes' theorem, the Poisson distribution, and Venn diagrams, all with interesting descriptions of the original contexts for their analysis. Even more interesting, though, are the chapters on applications. Insurance, the origins of statistics, medical decisions, the criminal justice system, weather forecasting and chaos, game theory, and information theory each get their own chapters.

The main recurring mathematical theme is Bayes' theorem; utility functions also occur often, but don't get as formal of a treatment. Surprisingly, Bayes' theorem arguments are never presented as a table, though I have found that many students prefer tables when doing Bayes' theorem problems.

I very much enjoyed this book for its insights into the history and personalities that led to the development of probability and statistics. While many probability textbooks give snippets of history and short biographies along with the formulas and examples, here the priorities are reversed: the story comes first, with formulas included if they are needed. It is a mixture of two genres: math history for general audiences (e.g. focusing on zero, pi, or e), and quantitative literacy (e.g. A Mathematician Reads the Newspaper by Paulos). However, its style can be rather florid at times, almost like Berlinski's A Tour of the Calculus. I found myself reaching for a dictionary more than once. But perhaps that is part of the horizon-expanding fun. The book also has many quotations from poets and philosophers that were unfamiliar to me; these perhaps could be placed in better context, by giving the relevant country and year.

No book is perfect, and there are a few mistakes in Chances Are…  that are common among my sophomore probability students. Some of these relate to terminology: the ideas of independent and mutually exclusive are mixed up (p. 52), the word "expected" is used in ways that conflict with technical usage (p. 44-45), and "inverse" is used where "complement" or "opposite" would be better (p. 20). Some formulas are incorrect as well: the expected value formula is presented as E= p(X) * A, where A is the amount promised to winners. No mention is made of the penalty for losing. The probability mass function given for a Poisson distribution has a term in the denominator that should be in the numerator. A drawing of a bell curve along with row 14 from Pascal's triangle has non-uniform x and y axes. Most of these occur while basic probability ideas are being discussed; once the real applications start, things get better.

For a book on the relationship between math and society, "Chances Are..." sometimes ignores some facts of modern life. For example, scientists and engineers are not mentioned in a list of "people who use probability today." Elsewhere (p. 38), the book says that a probability of 0.99874 is "as close to certainty as mortals know." This may have been true in de Moivre's day, but if we apply that probability to the over 30000 airline flights per day in the U.S. alone, we would average dozens of accidents each day. We could also have an interesting discussion about "Groups of simultaneous linear equations can be solved; non-linear ones usually cannot." (p. 215)

Even with those caveats, Chances Are…  deserves a place on the supplemental reading list for probability and statistics courses at many schools.

Andrew M. Ross was an undergraduate math major, got a PhD in operations research, currently teaches in an industrial engineering department, and is about to switch to a math department.

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