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Fat Chance: Probability from 0 to 1

Benedict Gross, Joe Harris, and Emily Riehl
Cambridge University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Thomas Hagedorn
, on
Fat Chance: Probability from 0 to 1 is a lively, reader-friendly introduction to introductory combinatorics, probability, and statistics.  The book is based on a General Education course created for non-math majors at Harvard College, then developed as an online course for HarvardX, and currently offered on the EdX online platform. All three authors are distinguished mathematicians and skilled expositors of mathematics, and the book is a pleasure to read.  
The book is an excellent text for anyone looking for a very enjoyable introduction to probability.  It can be read on its own or used as a textbook for a math course for non-math majors.  For students doing self-study, the accompanying HarvardX/EdX Fat Chance course is highly recommended as the video lectures closely (though not always) mirror the text’s presentation and organization. As of August, 2021, the online course is free to audit.  
The text is aimed at an audience without prior knowledge of probability and the only formal prerequisites are familiarity and proficiency with high school algebra.  The authors explicitly ask “the reader to be prepared to approach the mathematics with a spirit of adventure and exploration, with the understanding that, while some work will be required, the experience at the end will be worth it.”  The book has an informal, conversational style.  It is problem-based, with many examples discussed and worked out, with the goal being to teach the reader how to apply the underlying concepts rather than knowing specific formulas.  The authors do a nice job of maintaining mathematical precision, coaxing the reader along with comments on what to expect, such as,  “Look: this is a math book. We’re trying to pretend it isn’t, but it is.  That means that it’ll have jargon – we’ll try to keep it to a minimum.” 
The book is organized into three parts.  Part I: Counting consists of seven chapters focusing on counting techniques.  By the fifth chapter, which ends with a discussion of the birthday problem (The result that in any group of 23 or more people, two people will likely share the same birthday), readers will know how to calculate the probability of being dealt of a specific type of poker or bridge hand.  An interlude of two chapters (not needed for the rest of the book) discusses Pascal’s triangle, the binomial theorem, recursive sequences, and calculates the formula for the nth Catalan number.  Nicely, this part ends with a section (§7.7)  discussing both the useful applications of probability theory and the intrinsic beauty and fascination of the subject. 
Part II: Probability further develops the probability theory coming from simple counting problems and focuses on situations where multiple outcomes are possible.  Most often, the applications involve games, but applications to medical decision making and election results are also discussed.  Part II introduces the concepts of expected value, conditional probability, independence, and Bayes Theorem (shown to be useful in determining your best course of action when the Zombie Apocalypse occurs!).  It also covers Bernoulli trials and a delightful discussion of the Gambler’s ruin problem (random walks), illustrating the perils of trying to avoid the pitfalls of gambling with roulette.  Part II ends with a discussion of geometric probability.  It introduces problems involving continuously changing quantities and illustrates how to calculate the probability of seeing a friend in a cafeteria and the probability that a customer in a post office will need to wait (an introduction to queueing theory).
Probability at Large, Part III of the book, demonstrates the connection of probability with statistics.  The authors study what happens when one plays a game many times.  The ensuing probabilities can become too complex to calculate and require new tools for their calculation.  In Chapter 12, the expected value and variance of a game, the normalized form of a game, and the expected value and variance when games are added together are all discussed.  In Chapter 13, the normal distribution is introduced and used to calculate the probability of getting a certain number of heads when a coin is flipped 1000 times.  The authors then discuss how the same calculations can be used to calculate the probability distribution for any game played many times and conclude by discussing applications to election polling.   The book’s last chapter concludes with a discussion of examples in which probability can be misleading, such as when false positives occur in medical tests.  
Overall, this book is a very enjoyable introduction to probability and statistics.  It lets students see the larger connections in the subject without getting bogged down in the details of calculations and theory.  While the book is based upon a course developed for non-math majors, all students, including math majors, would profit by reading this book to learn basic probability and statistics. 
I note one typographical correction: On page 174, the value of e is listed incorrectly as 2.7128 instead of 2.71828.
About the reviewer:  Tom Hagedorn is Professor of Mathematics and Statistics at The College of New Jersey.  His mathematical interests are in number theory and abstract algebra.