When I first joined the mathematics department at Iowa State University, I was asked to redesign the syllabi for our two “quantitative literacy” courses, math 104 and math 105. These large lecture classes (150+ students per class) were designed to allow students to satisfy their mathematics graduation requirement without having to take something like calculus. The students in the class were characterized by a deep and abiding desire not to be there, and by their hatred or fear (or both) of mathematics in general. For this reason, although the courses are intended to be freshman-level, the audience often consists of upper-level students who have put the course off until the very last minute, and therefore have had three years or so to forget what little mathematics they learned in high school.

Math 104, as it existed at the time, was essentially a finite mathematics course, including some basic combinatorics and matrix theory. It appeared to me that most students found it unsatisfying to learn how to evaluate binomial coefficients or multiply matrices without having any idea at all as to why anybody would want to do these things. As I set about redesigning it, I was motivated primarily by a desire to make the course “real world relevant”, grounded in some unifying topic that students might have some inherent interest in. I decided to teach basic probability theory, with a focus on its applications to games and gambling. (I also threw in some illustrations of probability used in court as well.)

For the first few times I taught the course I used books like Gould’s *Mathematics of Games, Sports and Gambling *and Packel’s *The Mathematics of Games and Gambling*. Both are very good books, but neither worked well in the class, primarily because the students simply were not equipped to sit down and read a mathematics book that wasn’t written in four colors, with cartoons and colored boxes, and containing the other trappings of a high school text.

The book now under review is similar in tone and coverage to these two books, but has some nice features that they lack. This book, like those of Packel and Gould, teaches elementary probability and livens it up with extensive discussions of applications to various casino games. Specifically, after an introductory chapter which describes some basic mathematics and gives a short but interesting overview of the history of gambling, there are three chapters discussing basic probability theory, covering topics such as introductory combinatorics, compound events, conditional probability, expected value, and the binomial distribution.

Throughout these chapters there are lots of interesting examples, all relating to gambling games: roulette, poker, lotteries, keno, craps, bingo, sports betting and others. The examples given here are so numerous and diverse that it might have been helpful to assemble, at the back of the book, an index of all of them. (I should mention at this point that the author has a more recent CRC Press textbook, unseen by me, titled *Mathematics of Keno and Lotteries* that focuses on these two games.)

After these chapters, there are three additional ones that use the previously-developed probability material to focus on additional gambling issues. Chapter 5 is on modified casino gambling and discusses variations of some of the games discussed previously. Chapter 6 is an entire chapter on blackjack, which among other things contains an excellent discussion of card counting. And finally, chapter 7 is on betting strategies. Strategies for (among others) roulette, the lottery, and blackjack are discussed.

One minor quibble that I have with the selection of topics is that I think the author missed a couple of opportunities to present interesting examples that, I have found, students seem to enjoy. There is no discussion of the famous counter-intuitive birthday problem, for example, and also no discussion of the famous Monty Hall problem and its interesting history. Also, when I teach math 104, I like to discuss how expected value plays an important role outside the realm of gambling — in situations involving insurance company life insurance premium pricing, for example, or an airline company’s decision whether to overbook flights. But these are minor issues, easily addressed in class by an instructor who wishes to discuss them.

As mentioned earlier, this book has some advantages over those of Packel and Gould. One of the weaknesses of Packel’s book as a text was the relatively small number of exercises, particularly the kind of drill problems that students at this level need. Bollman’s book, by contrast, has lots of exercises. (Solutions to the odd-numbered ones appear in the book, and the publisher offers a password-protected complete solutions manual to instructors.)

A problem I noted with Gould’s book, at least as far as my course was concerned, was the fact that probability and gambling only occupied a fairly small portion of the book; there were also chapters on card tricks, statistics and game theory that were inappropriate for my course, at least as I designed it. This problem does not exist with Bollman’s book, all of which is devoted to the general topic of basic probability with applications to gambling and games.

Bollman’s book has other good features as well: it is very clearly written, and there are tons of examples, with detailed explanations. There are also lots of photographs and drawings that should help stimulate student interest. Student interest should also be enhanced by the plethora of stories about individual casinos and gambling events. And it is always nice to read a book by an author with a sense of humor. (Example: “The replacement of baccarat by Texas Hold‘Em in the 2006 version of *Casino Royale* represents something of a betrayal of the Bond legend even as it tapped into a wave of poker popularity.”)

Despite all these good features, I must regretfully say that I doubt this book would be successful as a math 104 text. Although the author writes clearly, he does cover things at a more sophisticated level than I think is warranted for a low-level, large lecture class full of people with little interest in investing any real time and effort in the course. He defines probability axiomatically, for example, and then commits the cardinal sin (for a class like this one) of actually stating theorems and proofs. Despite the fact that the theorems are the most basic ones and the proofs extremely easy and carefully written up, the mere mention of words like “theorem” and “proof” would likely cause my class to come to a screeching halt. I take the easy way out and assume the sample space is finite and consists of equally likely outcomes. This simplifying assumption, of course, allows a much easier definition of the probability of an event (which the author gives). Bollman also uses mathematical notation such as the sigma sign for summations, and also omits the trappings, described earlier, of a more elementary text.

Although I don’t believe that this book could be used for a course like math 104, I do believe that it can be very successfully used in other contexts and for other kinds of courses. It would, for example, be a great text for a seminar (especially an honors seminar). Also, an instructor teaching other courses (a majors course in probability, an introductory course in combinatorics, or even something similar to math 104) would find this book a treasure trove of fascinating examples. (I’m scheduled to teach math 104 in the fall, and intend to have this book close at hand when preparing lectures.) Finally, this book might be useful in a non-academic setting as well; I think that people interested in gambling would find large portions of it interesting.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.