The Theory of Gambling and Statistical Logic

In the foreword to this second edition, Richard A. Epstein notes that on the publication of the revised 1977 edition, he presumed that “the fundamentals of gambling theory had been fully accounted for.” Computers, of course, have changed that; as he rightly concedes. Nonetheless, quick access to nearly unlimited amounts of data from gambling simulations has not diminished the role of theoretical probability in analyzing games of chance, and Epstein’s revised work continues to excel in bringing mathematical rigor to questions arising from gambling. While it’s certainly possible to do good gambling mathematics using nothing more than advanced arithmetic and some simple combinatorics, the full power of integrals and iterated summations brings a lot to this subject, and the result is a comprehensive survey of gambling mathematics — and the mathematics of games in general.

The standard casino games are given good treatment, as are a variety of games where game theory is a more appropriate mathematical tool than probability. Unusual in this work is a chapter-length treatment of Parrondo’s Principle on the combination of two negative-expectation games to yield a positive expectation. The “statistical logic” mentioned in the title refers to more serious applications of probability as it is used in stock market analysis and sports tournaments. Bits of psychology and philosophy are included in a discussion of the statistical evidence for ESP and common fallacies associated with risk; these serve as a reminder that mathematical precision is not the only factor by which games — of multiple sorts — are assessed.     

Taken together, this is a first-rate survey of gambling mathematics that deserves a place in the library of anyone interested in the numbers that underlie that industry.

Mark Bollman (mbollman@albion.edu) is associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. Mark’s claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.