Problem: 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?
My solution: the event that the series lasts exactly five games is a composition event of A wins 4 games in a series of 5 games and B or B wins 4 games in a series of 5 games.
P(exactly 5 games)
= 5C4.p^4.(1-p) + 5C4.p.(1-p)^4
= 5[p^4.(1-p) + p.(1-p)^4]
However, the book solution is: 4[p^4.(1-p) + p.(1-p)^4]
Is my solution wrong?
Best Answer
The terminating condition is that one team has its fourth win; no more games happen after this. The favoured event is that this happens on the fifth game.
So you want the event that either:
Hence $\binom{4}{3} p^3(1-p)p + \binom{4}{1}p(1-p)^3(1-p)\\ = 4 p^4(1-p)+4 p(1-p)^4$