Chapter 8.3, Problem 32E

Tennis Consider a game of tennis between player A and player B that has reached the state deuce (or 40–40). Continuing from this point, after each rally, the game will be in one of the five  states—-A wins, B wins, Deuce, Advantage A, or Advantage B. Suppose that the stochastic matrix for the remainder of the game is

$\begin{array}{l}\underset{\text{wins}}{\text{A}}\underset{\text{wins}}{\text{B}}\underset{\text{D}}{}\underset{\text{A}}{\text{Ad}}\underset{\text{B}}{\text{Ad}}\\ \begin{array}{c}\text{A wins}\\ \text{B wins}\\ \text{Deuce}\\ \text{Ad A}\\ \text{Ad B}\end{array}\left[\begin{array}{ccccc}1& 0& 0& .4& 0\\ 0& 1& 0& 0& .6\\ 0& 0& 0& .6& .4\\ 0& 0& .4& 0& 0\\ 0& 0& .6& 0& 0\end{array}\right]\end{array}$.

a. What is the probability that player A will win the game? Player B?

b. What is the expected number of rallies?