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Fibonacci and Catalan Numbers: An Introduction by Ralph P. Grimaldi

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Chapter 28

The Catalan Numbers at Sporting Events

This chapter is novel in that it takes us from our typical mathematical applications to one involving probability concepts in conjunction with well-known sporting events.

Example 28.1: The following example is due to Louis Shapiro and Wallace Hamilton. The discussion given here follows that presented in Reference [35].

Given a positive integer n, consider a series of at least n games but no more than 2n − 1 games, where the winner is the first to win n of the games played. For instance, at a grand slam (such as Wimbledon or the U. S. Open), the women's final is won by the first woman to win two of the three (possible) sets. So in this case, n = 2. When the men play the final at such a grand slam, the champion is the first to win three of the five (possible) sets. This time n = 3.

Each October, the American and National League pennant winners square off for the world series—or October classic. For this series, n = 4 and the first team to win four of the seven possible games is the world champion (for that year).

Let A and B denote the two opponents for the three preceding situations. Assume the probability opponent A wins a given game (or set) is p; for opponent B, the probability is then 1 − p = q. Now we shall let En denote the expected number of games played, if the first opponent to win n of at most 2n − 1 games (or sets) wins the championship. It can be shown that

We confirm the result for E3 as follows. In this case,

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