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Introduction to Probability with Texas Hold'em Examples

Frederic Paik Schoenberg
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
BLL Rating: 

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

[Reviewed by
Mark Bollman
, on

I like a book that is what it says it is, and Introduction to Probability with Texas Hold’em Examples is a fine example. The title says it all: this book treats the standard topics of elementary probability, all of them illustrated with examples drawn from Texas Hold’em poker. The mathematics is appropriately rigorous, complete with improper integrals, density functions, and limits everywhere that they should be — it is the laserlike focus of the examples and exercises that sets this book apart from other probability textbooks at this level.

One weakness of this choice is that some fundamental topics of probability — for example, permutations and independent events — aren’t as amenable to a Texas Hold’em approach. Nonetheless, the commitment to drawing examples from this mathematically rich game is impressive. When introducing Bayes’ Theorem, the classical example of medical testing is used, but this is an extremely rare exception. The book is incredibly well-researched — examples are drawn from actual televised poker games, and many explorations of the probabilities in play in a given game situation conclude with a sentence about what really happened, which is a nice touch.

That said, this is not a book from which to learn Texas Hold’em, and the author makes that point clear in the first paragraph of the preface. While there’s a quick run-through of the rules in an appendix and an invaluable glossary of important terms in another, the intended reader of this book is someone already knowledgeable about the game and who is looking for insight into the mathematics behind it. At the same time, this is as good a resource as exists for a reader in the reverse situation: someone looking for or interested in a collection of applications of known mathematics to Texas Hold’em.

Mark Bollman ( is associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His 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.

Probability Basics
Meaning of Probability
Basic Terminology
Axioms of Probability
Venn Diagrams
General Addition Rule

Counting Problems
Sample Spaces with Equally Probable Events
Multiplicative Counting Rule

Conditional Probability and Independence
Conditional Probability
Multiplication Rules
Bayes’ Rule and Structured Hand Analysis

Expected Value and Variance
Cumulative Distribution Function and Probability Mass Function
Expected Value
Pot Odds
Luck and Skill in Texas Hold’em
Variance and Standard Deviation
Markov and Chebyshev Inequalities
Moment Generating Functions

Discrete Random Variables
Bernoulli Random Variables
Binomial Random Variables
Geometric Random Variables
Negative Binomial Random Variables
Poisson Random Variables

Continuous Random Variables
Probability Density Functions
Expected Value, Variance, and Standard Deviation
Uniform Random Variables
Exponential Random Variables
Normal Random Variables
Pareto Random Variables
Continuous Prior and Posterior Distributions

Collections of Random Variables
Expected Value and Variance of Sums of Random Variables
Conditional Expectation
Laws of Large Numbers and the Fundamental Theorem of Poker
Central Limit Theorem
Confidence Intervals for the Sample Mean
Random Walks

Simulation and Approximation Using Computers

Appendix A: Abbreviated Rules of Texas Hold’em
Appendix B: Glossary of Poker Terms
Appendix C: Solutions to Selected Odd-Numbered Exercises

References and Suggested Reading