Chapter 4, Problem 4.27STPE

Two teams will play a series of games, with the winner being the first team to win a total of 4 games. Suppose that, independently of earlier results, team 1 wins each game it plays with probability $0<p<1$ .Let N denote the number of games that are played.

a. Show that $P\left(N=6\right)\ge P\left(N=7\right)$ with equality only when $p=\frac{1}{2}$.

b. Give an intuitive explanation for why equality results when $p=\frac{1}{2}$.

Hint: Consider what needs to be true in order for the number of games to be either 6 or 7.

c. If $p=\frac{1}{2}$, find the probability that the team that wins the first game wins the series.