For the second time this year, a book dealing with the application of the mathematical sciences to baseball has crossed my desk. The first was

Teaching Statistics Using Baseball, by Jim Albert, reviewed earlier in

*Read This!*. Now we have

*A Mathematician at the Ballpark*, by Ken Ross. This book emphasizes probability rather than statistics, conversational exposition rather than case studies, and gentle suggestions and open questions rather than homework exercises. Albert's book stays true to the mission of exploring baseball, using statistical techniques. He examines questions such as, "How certain should you be of stealing a base before you attempt it?" and "Is a sacrifice bunt worth the sacrifice?" While several of Ross's chapters have titles like, "Who's the Best Hitter?" and "Will the Yankess Win if Steinbrenner is Gone?" the questions often go unanswered. Instead, Ross's book is devoted to conveying concepts of probability, and goes beyond baseball to involve roulette, basketball, bowling, and medical studies to do so.

As an informal introduction to the concepts of probability which makes liberal (but not exclusive) use of baseball examples, this book has much to recommend it. Ross begins his book by examining some of the statistics put forth by statisticians as being more reflective of hitting performance than the standard batting average (AVG) and runs batted in (RBI), including informal discussion as to why the proferred statistics are more useful than the standard ones. An engaging discussion of the streakiness of hitters is given, involving both extremes (many short streaks versus fewer, longer ones) and DiMaggio's 56-game hitting streak. The concepts of expected value and odds are both introduced in the context of betting on baseball games. In fact, so much of the exposition centers on gambling (usually on baseball games) that the book might have been called *A Mathematician Goes to the Sports Book in Vegas*.

The time spent on betting on baseball rather than baseball itself is not necessarily a bad thing. Indeed, the greatest strength of the book is how Ross handles odds. In many introductory statistics texts and liberal arts mathematics books, the section on odds is a short addendum to the section introducing probability, in which the relationship

Odds against A = a:b *if and only if* P(A loses} = a/(a + b} is introduced. Students get some practice finding odds given a probability, finding a probability given the odds, and are introduced to a real-world example (usually horse racing) where odds are used rather than probability. And that's it. But Ross goes further, finding situations where presenting the odds makes things simpler than presenting the corresponding probabilities, such as describing your chances of doubling your betting budget before losing it, and shows how oddsmakers ensure a profit for the house when they don't know the true odds of a team winning a game. Ross even puts this information to use by developing and experimenting with a scheme to beat the oddsmakers. This makes the book a welcome addition to the toolbox of a probability instructor. As an exploration of the game of baseball using probability, though, the book falls short. Ross has not really explored the question, "What can mathematics teach us about baseball?"

There has been a recent spike of interest in the application of rigorous mathematics to the study of baseball, primarily due to the Society of American Baseball Research (SABR), a group of sportswriters, statisticians, and just plain fans who set out to replace conventional wisdom, hunches and intuition with rigorous and verifiable mathematical analysis. (For further information, see http://www.sabr.org.) They, in turn, owe their existence to the availability of data sets on the web, and to statistical software packages capable of handling the data. Part of their appeal is that statistically savvy readers can verify results and explore questions on their own. Ross doesn't mention SABR until the appendices, where he gives brief summaries of several provocatively titled articles they have published. He does not mention the on-line availability of data sources, or take much advantage of them himself to explore common situations in baseball. For example, despite the abundance of situations in a baseball game that involve conditional probabilities (batting average given the pitcher is a leftie, probability a team wins the World Series if they've got home-field advantage), Ross uses a medical example for conditional probability because of the limitations on the accuracy of a baseball example. The emphasis on the concept overriddes its possible application to the game.

But perhaps that's the point. This is, above all, a probability book, and the situations mentioned above are primarily statistical. All of the standard topics from introductory probability are here, from the probabilities of all possible outcomes summing to 1, to Simpson's paradox. Ross obviously loves baseball; anyone who claims to be a lifelong fan of the Eugene Emeralds isn't in it for the glory of a World Series victory. But it's his love for probability that comes through most strongly. *A Mathematician at the Ballpark* is a tidy package of interesting examples and engaging discussions, which, by and large, emphasize the mathematics, rather than shed new insight on the game.

Baseball fans (and others) looking for some help understanding concepts of probability through informal discussions and nonstandard examples need look no further than this book. Those looking for information on what the mathematical sciences can say about the national pastime would do well to consider Teaching Statistics using Baseball. Jim Albert has also written, with Jay Bennett, the book Curve Ball, which is aimed at a more general audience, and may be a good choice for someone interested in seeing different ways of using mathematics to analyze the game of baseball in a less structured manner.