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The Doctrine of Chances: Probabilistic Aspects of Gambling

Stewart N. Ethier
Publication Date: 
Number of Pages: 
Probability and Its Applications
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Peter Rabinovitch
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I fell in love with probability theory when I read Patrick Billingsley’s 1983 expository paper “The Singular Function of Bold Play.” This paper explained how to decide what amount to bet on an unfavorable game so as to maximize the probability of winning a fixed sum. Contrary to my intuition at the time, the answer is not to make many small bets, but rather stake a lot of the money on the first bet, hence the name “bold play.’ Of course there are many details I just left unspecified, but regardless it is a fascinating result.

Flash forward eight years. I’m sitting in a casino in the Bahamas, in front of a five cent slot machine, pushing buttons and imagining random walk with negative drift, representing how much money I was going to go back to the room with. Every now and then, the plot would jump up a little, sometimes more than a little, almost enough to convince me that if played just a little longer, I’d win enough money to extend our vacation. But I knew better… the machine would win.

Nineteen years later, I’m very happy to be reviewing Ethier’s labor of love The Doctrine of Chances: Probabilistic Aspects of Gambling.

The book is split into two sections. The first section has chapters on the mathematical background needed to understand the second section, which has chapters about ten casino games.

The mathematical chapters cover probability, including martingales and Markov chains, some game theory, and a few topics more closely related to gambling, such as gambler’s ruin, betting systems and bold play. The material in this section is very nicely done, with many interesting, and some challenging exercises. This part of the book has few proofs of the standard results. Some are illustrated via examples, some are simply stated, and some are included exercises. Although the reader does not need measure theoretic probability to understand these sections, I do believe that the reader would need a corresponding level of mathematical sophistication, and a good probability text at their side to benefit the most from these chapters. On the other hand, results that are less standard, and more specific to gambling, are generally included.

The chapters on the casino games include all the favorites: slot machines, roulette, black jack, and craps as well as some less familiar ones, such as keno, faro and 007’s favorite, baccarat. Each chapter explains the rules of the game, and presents analysis of specific playing strategies, many with detailed calculations, tables, etc. Have a look at the sample chapter on video poker at the Springer web site to get a better idea of how each game is analyzed.

Each chapter ends with many exercises and fascinating historical notes and the book ends with a few mathematical appendices.

There are many other books that attempt to cover similar material at varying levels of rigour, perhaps the closest being Richard Epstein’s The Theory of Gambling and Statistical Logic. Ethier’s book is clearly aimed at a more mathematically sophisticated audience than Epstein’s, and for that reason I enjoyed Ethier’s much more.

The main critique I can make is that there are almost no graphs in the book, and a topic such as gambling, with ruin problems, random walks, arcsine laws, etc. is to me a very visual subject. If such images had been included it would have made the book just that much more understandable and enjoyable.

If you have the right background, this is the book to read to understand that you will never become rich at the casino. But at least you will love learning why.

Peter Rabinovitch is a Systems Architect at Research in Motion, and a PhD student in probability. When not working, he likes to eat spicy food. He does not enjoy losing money at casinos.


Preface.- Part I Theory. 1. Review of Probability. 2. Conditional Expectation. 3. Martingales. 4. Markov Chains. 5. Game Theory. 6. House Advantage. 7. Gambler’s Ruin. 8. Betting Systems. 9. Bold Play. 10. Optimal Proportional Play. 11. Card Theory.- Part II Applications. 12. Slot Machines. 13. Roulette. 14. Keno. 15. Craps. 17. Video Poke. 18. Faro. 19. Baccarat. 20. Trente et Quarante. 21. Twenty-One. 22. Poker.- A Appendix. A.1. Results from algebra and number theory. A.2.Results from analysis and probability.- List of Notations. Answers to Selected Problems. References.- Index.