Gambling games have inspired two major mathematical developments related to our work. In the seventeenth century, a writer named the Chevalier de Méré posed a question about probabilities with repeated rolls of the dice to the eminent mathematician and philosopher Blaise de Pascal. The problem led Pascal and another legend of mathematics, Pierre de Fermat, to develop the foundations for a modern theory of probability. In the twentieth century, the game of Poker with its complex mix of chance and strategy became a central problem for the development of game theory. Both Emilé Borel and John von Neumann published early work on the game in restricted settings. The psychology of betting and bluffing in Poker became a useful metaphor for the variables at play in economic and political interactions. While the complexity of Poker has proven too great to admit anything like a complete mathematical theory, working algorithms for playing the game can be extremely lucrative.

In this chapter, we discuss two popular gambling games: Craps and Poker. Our analysis of these games will motivate several new ideas. For the game of Craps, we explain the technique of “introducing a conditional,” which allows us to compute a probability in an infinite game. Straight Poker, in turn, presents the problem of counting card hands. We apply the Combinations Formula (Theorem 11.12) to compute the probability of Poker hands. We then return to the game Five-Card Stud and in particular address Question 1.2 from Chapter 1. Our resolution of this question will motivate Bayes' Theorem (Theorem 14.5).

Craps

The dice game Craps creates excitement from the simple of act of repeated rolling of a pair of dice. We introduce this casino game now.