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Two teams A and B play a series of games until one team wins four games. We assume that the games are played independently and that the probability that A wins any game is p. What is the probability that the series lasts exactly five games?

Feb 28, 2020

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$$P[\text{5 games}] = P[\text{5 games | A wins}]+P[\text{5 games | B wins}]$$

$$P[\text{5 games | A wins}]=p^4 (1-p)\\ P[\text{5 games | B wins}]=p (1-p)^4\\ P\text{5 games}=p^4(1-p)+p(1-p)^4 = \\ p(1-p)[p^3+(1-p)^3]$$

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Feb 28, 2020