Mathematics of Keno and Lotteries

This book is a sequel of sorts to Bollman’s earlier Basic Gambling Mathematics: The Numbers Behind the Neon (hereafter denoted BGM). This earlier book looked at basic discrete probability with reference to a wide variety of games of chance, including poker, blackjack, roulette, craps, and, yes, keno and lotteries. The book now under review focuses on these last two games and discusses them in somewhat more detail than in BGM. Parts of BGM are reproduced here, but there is a great deal of new material.

While most people are probably familiar with the general concept of a lottery, not everybody may be as conversant with the mechanics of keno, so perhaps a brief summary at the outset is appropriate. In its most basic form, a gambler buys a card with the numbers 1 through 80 on it, and selects n of these numbers, where n can be any number between 1 and 20. (The gambler generally gets to choose the value of n, thereby creating a “pick n” game, and the casino has different payout schedules depending on that choice.) The casino then selects 20 numbers at random, and the payoff to the gambler depends on how many of his numbers are among the ones selected by the casino (i.e., how many numbers he has “caught”). There are many variations on this theme, of course, and most of them are discussed in this text.

There are four chapters in the book, the first of which is introductory. The two games referred to in the title are introduced and a relatively brief but informative history of each of them is provided. We learn, for example, that keno had its historic roots in China, about 1500 years ago, and that the first people to use lotteries as a form of recreation were the ancient Romans. There is also a nice discussion of the old-time “Numbers racket” in the United States.

Chapter 2 is a quick (roughly 40 page) summary of the concepts of discrete probability that will be used in the text. Topics covered include the definition and basic facts about probability, introductory combinatorics, random variables and expected value, and the binomial distribution. In contrast to BGM, where probability is introduced via axioms, here a simplified approach is taken under the assumption that all sample spaces are finite with equally likely events. (Purists might object to the fact that, before the author even states what probability is, he states, as an unproved theorem, that the probability of the complement of an event A is equal to 1 – P(A).)

Occasional reference is made in chapter 2 to games other than keno, such as roulette, but none of these other games are discussed in any real detail. Most of the illustrations of the concepts discussed in this chapter involve keno and lotteries. One interesting question posed by the author, that I had never thought to ask before, is: how do you compute the expected value of a lottery when one of the prizes is a free lottery ticket, rather than a cash payout?

Chapters 3 and 4 look at keno and lotteries, respectively, in more detail. A lot of time is spent discussing mathematical aspects of variations of these games that are found in various casinos and state lotteries; the author also discusses, in both chapters, a number of strategies for winning at keno and the lottery and explains their weaknesses.

The book is filled with dozens of references to various specific casinos and the games they play (“In March 2016, the Alamo Casino was offering 740 for 1 on a catch-5 event…”), and also references to many state lotteries — so many such references, in fact, that I often found myself wondering just how the author has managed to accumulate so much data. I suspect his knowledge of specific casinos and their games is based in large part on personal visitation, since on occasion he even describes the physical layout of a casino:

The Alamo is not a casino resort, but a truck stop with fast food restaurants and a gas station. The casino is small, with two blackjack tables and a sports book among the slot and video poker machines. As the casino is not the primary focus, offering competitive odds is less of a concern for management.

In addition, at several points in the text the author compares and contrasts the house advantage for various casinos on the same game. He notes, for example, that with regard to the pick-5 keno game, the Treasure Island casino offers better payoffs than the El Cortez casino, and points out that “This result stands as a counterexample to the general rule that gambling conditions are typically more favorable to players in downtown Las Vegas than on the Las Vegas Strip.”

I found details like these to be interesting and amusing, and I suspect students will too. The author did, however, miss one of my favorite lottery stories: in 1980, a woman named Maureen Wilcox bought tickets for both the Massachusetts and Rhode Island state lotteries. She picked winning numbers on both tickets — but, unfortunately, on her Rhode Island ticket, she picked the winning numbers for the Massachusetts lottery, and vice-versa.

The discussions in these two chapters are sufficiently wide-ranging that mathematics other than combinatorics and basic discrete probability is discussed. For example, the author develops some basic financial mathematics in chapter 4 to discuss annuities as a form of lottery payment. Sometimes the author even ventures outside the realm of mathematics, as in his discussion of the tax implications of gambling winnings on page 73.

The writing style, as in BGM, is generally quite clear and inviting. Most of the book should be accessible to anybody with a basic knowledge of high-school mathematics, but a reader should be prepared to think carefully about what he or she is reading. The introduction to discrete probability concepts is somewhat more condensed than in BGM, for example, and so perhaps some prior exposure to these ideas would be useful for a reader. Some mathematical symbolism, such as the sigma notation for summation, sometimes gets used, and there is an occasional reference to things like Mathematica.

Each chapter ends with a selection of exercises. In a good compromise, the author has made numerical solutions to many of them available at the end of the book, so that a student can check his or her work for correctness, but has not included detailed solution methods, so that an instructor can still feel comfortable in assigning these problems for graded homework submission. The publisher has made available a password-protected solutions manual for faculty members. This manual is 18 pages long and contains solutions, both numerical answers and brief explanations, for the vast majority, perhaps all, of the exercises in the book. Some of the exercises require the construction of an Excel spreadsheet.

I mentioned in my review of BGM that although I thought that book would likely be too demanding for a truly low-level “mathematics literacy” course in probability populated by a captive audience of students who didn’t really want to be there, it would make a good text for a seminar with more motivated students. The same comments apply to this book, although I think that the variety of gambling games discussed in BGM makes that book somewhat more versatile. Despite its narrower focus, however, this book is a very useful addition to the elementary literature on probability and gambling and should, like BGM, also make an excellent reference and source of interesting examples for an instructor teaching a course in combinatorics or probability.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.