Competition problem — Combinatorics

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Combinatorics Difficulty 5.2 AIME, harder Prove it United States

Problem:

The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game $n$ is $\frac{n-1}{6}$ . What is the probability that the Red Sox will win the series?

Solution

Solution:

Note that if we imagine that the series always continues to seven games even after one team has won four, this will never change the winner of the series. Notice also that the probability that the Red Sox will win Game $n$ is precisely the probability that the Yankees will win Game $8-n$ . Therefore, the probability that the Yankees win at least four games is the same as the probability that the Red Sox win at least four games, namely $1/2$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.