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