This book introduces and develops some of the important and beautiful elementary mathematics needed for rational analysis of various gambling and game activities. It assumes the reader has a solid background in secondary school mathematics. Mathematical ideas are interspersed with applications to casino games and other gambling and gaming activities, with emphasis on the mathematical ideas behind games and playing them strategically. The book includes many helpful internet links that give additional information about games and allow the reader to play them and demonstrate them online.

### Table of Contents

Preface to the First Edition

Preface to the Second Edition

1. The Phenomenon of Gambling

2. Finite Probabilities and Great Expectations

3. Backgammon and Other Dice Diversions

4. Permutations, Combinations, and Applications

5. Play it Again Sam: The Binomial Distribution

6. Elementary Game Theory

7. Odds and Ends

Answers/Hints for Selected Exercises

Bibliography

Index

About the Author

### Excerpt: Ch. 5.4 Betting Systems (p. 90)

Despite a rational interpretation of probabilities, expectations, and even the binomial distribution, there are many gamblers, both casual and inveterate, who believe that they have found or will find a winning system of betting in games with a house edge. Most of these proposed systems are naive, based on superstition, humorous, or sometimes pathetic. There are certain systems, however, which are compelling, difficult to deflate, and perhaps even successful for specific gambling goals (lowering the house edge not being one of them). We will not consider systems based upon elaborate record keeping to spot non-random trends of the wheel, dice, or whatever. Most such efforts depend on after-the-fact reasoning and are doomed to failure, though cases of faulty (or crooked) apparatus have very occasionally led to success. We also dismiss systems for "knowing" what is due or for recognizing "hot" dice and "friendly" wheels until such time as parapsychology finds itself on a firmer foundation. The systems we do consider base their considerable appeal on varying the bet size in some fashion depending on what has happened previously. We analyze several such systems knowling full well that there are many other "sure-fire" systems not treated here.

A most intriguing system is the "double when you lose" or *Martingale* strategy. In this system one starts betting at a given stake, say 1 unit, and doubles the previous bet after a loss while returning to the original 1 unit stake after each win. It is easy to see that after *r* wins the player will be ahead by *r* units, and that the only thing the player needs to worry about is a long streak of consecutive losses. Indeed if a player goes to Las Vegas with the primary goal of coming out ahead (no matter by how little), it is hard to imagine a better system. If one's bankroll is $63 and a $1 minimum bet is in effect, one should play the doubling system starting with a $1 even payoff bet, planning to quit a go home immediately after winning for the first time. The only way one can fail is to lose the first 6 bets (1 + 2 + 4 + 8 + 16 + 32 = 63) which has probability (1 - *p*)^{6} (where *p* is, as usual, the probability of winning on each trial). Even for American roulette (*p* = 0.474) the probability of losing the first 6 bets is (0.526)^{6} = 0.021 ≈ 1/50, a comfortably small magnitude. It is true that the loss of $63 in this unlikely event will be much greater than the hoped for $1 gain, but is not the payoff (bragging to all one's friends about how you beat the Vegas syndicate) worth the small risk? The answer is "yes" if winning something (never mind the air fare) is the main goal. The answer is "no" if one believes that the mathematical expectation per dollar bet has been altered. The system would be foolproof but for two vital facts:

- The player has only a finite amount of capital.
- The casino imposes a maximum on any given bet.

Each of these facts imposes a limit upon the number of losses beyond which the doubling must be abandoned. Assuming that the doubling system can only be followed *n* times, let us apply the expectation concept to this "go home a winner" system. Then

*p*(losing on all *n* bets) = *q*^{n}

and hence

*p*(winning on one of of the first *n* bets) = 1 - *q*^{n}.

Since the net gain after the first win is 1 unit while the damage after *n* losses is 1 + 2 + 4 + 8 + ^{...} + 2^{n - 1} = 2^{n} - 1 units, we have

*X* = (1 - *q*^{n})(1) - *q*^{n}(2^{n} - 1) = 1 - *q*^{n}2^{n}.

In the special case *p* = 1/2, we have *q* = 1 - *p* = 1/2 and *X* = 0. Thus, as expected, the doubling strategy does not affect the expectation per dollar bet in this special case.

### About the Author

**Ed Packel** received his BA from Amherst College in 1963 and his PhD from MIT in 1967. He is currently Volwiler Professor of Mathematics at Lake Forest College, where he has taught mathematics and computer science courses since 1971.

Packel’s initial work in functional analysis resulted in several research articles and an introductory graduate text. Subsequent research has involved game theory, social choice theory and information-based complexity. He is also heavily involved in using technology, primarily *Mathematica*, as a tool for teaching and research in mathematics. This work has led to publication of a variety of articles, several Mathematica-related books and 15 years of *Rocky Mountain Mathematica* July workshops.

The author has a healthy interest in games and sports. His fascination with bridge and backgammon is complemented with a long-running monthly applied probability seminar (alias poker game). He has played and coached soccer, competed as a distance runner, and enjoys the game of golf.