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**The doctrine of chances. Probabilistic aspects of gambling.**
*(English)*
Zbl 1198.60001

Probability and its Applications. Berlin: Springer (ISBN 978-3-540-78782-2/hbk; 978-3-540-78783-9/ebook). xiv, 816 p. (2010).

This textbook on probabilistic aspects of gambling is directed to those already familiar with probability at the post-calculus, pre-measure-theory level. Gambling is a fitting application of probability theory on which to base such a study, both because of its prominent role in the historical development of the subject and because gambling is one of the few applications in which the probabilistic models are often exactly correct.

The book is in two parts of eleven chapters each. Part I, Theory, begins with a chapter on review of probability, followed by chapters on conditional expectations, martingales, Markov chains, game theory, house advantage, gambler’s ruin, betting systems, bold play, optimal proportional play and card theory (shuffling, dealing, card counting). Part II, Applications, has a chapter on each of slot machines, roulette, Keno, Craps, house-banked poker, video poker, Faro, Baccarat, Trente et Quarante, twenty-one, and poker. Each chapter has a good selection of problems (answers, but not solutions, available on author’s web page), and some interesting notes, including some very interesting history.

There is plenty of material here for a solid two-semester course, but there is enough independence among the chapters to allow for a variety of one-semester courses covering a subset of the chapters. The book is a welcome and well-researched addition to the field.

The book is in two parts of eleven chapters each. Part I, Theory, begins with a chapter on review of probability, followed by chapters on conditional expectations, martingales, Markov chains, game theory, house advantage, gambler’s ruin, betting systems, bold play, optimal proportional play and card theory (shuffling, dealing, card counting). Part II, Applications, has a chapter on each of slot machines, roulette, Keno, Craps, house-banked poker, video poker, Faro, Baccarat, Trente et Quarante, twenty-one, and poker. Each chapter has a good selection of problems (answers, but not solutions, available on author’s web page), and some interesting notes, including some very interesting history.

There is plenty of material here for a solid two-semester course, but there is enough independence among the chapters to allow for a variety of one-semester courses covering a subset of the chapters. The book is a welcome and well-researched addition to the field.

Reviewer: Gerald A. Heuer (Moorhead)