''The Mathematics of Games and Gambling'' is a book on

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

and its application to

games of chance A game of chance is in contrast with a game of skill. It is a game whose outcome is strongly influenced by some randomizing device. Common devices used include dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from ...

. It was written by Edward Packel, and published in 1981 by the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

as volume 28 of their New Mathematical Library series, with a second edition in 2006.

Topics

The book has seven chapters. Its first gives a survey of the history of gambling games in western culture, including brief biographies of two famous gamblers,

Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...

and

Fyodor Dostoevsky Fyodor Mikhailovich Dostoevsky (, ; rus, Фёдор Михайлович Достоевский, Fyódor Mikháylovich Dostoyévskiy, p=ˈfʲɵdər mʲɪˈxajləvʲɪdʑ dəstɐˈjefskʲɪj, a=ru-Dostoevsky.ogg, links=yes; 11 November 18219 ...

, and a review of the games of chance found in Dostoevsky's novel '' The Gambler''. The next four chapters introduce the basic concepts of probability theory, including expectation,

binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...

s and compound distributions, and

conditional probability In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occur ...

, through games including

roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...

keno Keno is a lottery-like gambling game often played at modern casinos, and also offered as a game in some lotteries. Players wager by choosing numbers ranging from 1 through (usually) 80. After all players make their wagers, 20 numbers (some va ...

craps Craps is a dice game in which players bet on the outcomes of the roll of a pair of dice. Players can wager money against each other (playing "street craps") or against a bank ("casino craps"). Because it requires little equipment, "street c ...

chuck-a-luck Chuck-a-luck, also known as birdcage, is a game of chance played with three dice. It is derived from grand hazard and both can be considered a variant of sic bo, which is a popular casino game, although chuck-a-luck is more of a carnival game ...

backgammon Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back nearly 5,000 years to the regions of Mesopotamia and Pe ...

, and

blackjack Blackjack (formerly Black Jack and Vingt-Un) is a casino banking game. The most widely played casino banking game in the world, it uses decks of 52 cards and descends from a global family of casino banking games known as Twenty-One. This fami ...

. The sixth chapter of the book moves from probability theory to

game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

, including material on

tic-tac-toe Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with ''X'' or ''O''. T ...

, matrix representations of

zero-sum game Zero-sum game is a mathematical representation in game theory and economic theory of a situation which involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is e ...

s, nonzero-sum games such as the

prisoner's dilemma The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...

, the concept of a

Nash equilibrium In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...

, game trees, and the

minimax Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When de ...

method used by computers to play two-player strategy games. A final chapter, "Odds and ends", includes analyses of bluffing in

poker Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game w ...

horse racing Horse racing is an equestrian performance sport, typically involving two or more horses ridden by jockeys (or sometimes driven without riders) over a set distance for competition. It is one of the most ancient of all sports, as its basic p ...

, and

lotteries A lottery is a form of gambling that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of ...

. The second edition adds material on

online gambling Online gambling is any kind of gambling conducted on the internet. This includes virtual poker, casinos and sports betting. The first online gambling venue opened to the general public was ticketing for the Liechtenstein International Lottery in ...

systems, casino poker machines, and

Texas hold 'em Texas hold 'em (also known as Texas holdem, hold 'em, and holdem) is one of the most popular variants of the card game of poker. Two cards, known as hole cards, are dealt face down to each player, and then five Community card poker, communit ...

poker. It also adds links to online versions of the games, and expands the material on game theory.

Audience and reception

The book is aimed at students, written for a general audience, and does not require any background in mathematics beyond high school algebra. However, many of its chapters include exercises, making it suitable for teaching high school or undergraduate-level courses using it. It is also suitable for readers interested in

recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

. Although it could also be used to improve readers' ability at games of chance, it is not intended for that, as its overall message is that gambling games are best avoided. Reviewer Sarah Boslaugh notes as a strength of a book the smooth interplay between its mathematical content and the context of the games it describes. Despite noting that the book's description of modern games is based on American practice, and doesn't address the way those games differ in Britain, reviewer Stephen Ainley calls the book "very enjoyable", adding that "it is hard to see how it could be done better or more readably". Reviewer J. Wade Davis calls it "accessible and very entertaining".

Recognition

The Basic Library List Committee of the

has listed this book as essential for inclusion in undergraduate mathematics libraries. It was the 1986 winner of the

Beckenbach Book Prize The Beckenbach Book Prize, formerly known as the Mathematical Association of America Book Prize, is awarded to authors of distinguished, innovative books that have been published by the Mathematical Association of America (MAA). The prize was esta ...

References

{{DEFAULTSORT:Mathematics of Games and Gambling, The Games of chance Probability theory Mathematics books 1981 non-fiction books 2006 non-fiction books