Chapter 10.5, Problem 27E

Exercises 27-30 concern the Markov chain model for scoring a tennis match described in Section 10.1, Exercise 31. Suppose that players A and B are playing a tennis match, that the probability that player A wins any point is p = .6, and that the game is currently at “deuce.”

27. How many more points will the tennis game be expected to last?

31. To win a game in tennis, one player must score four points and must also score at least two points more than his or her opponent. Thus if the two players have scored an equal number of points (four or more), which is called “deuce” in tennis jargon, one player must then score two points in a row to win the game. Suppose that players A and B are playing a game of tennis which is at deuce. If A wins the next point it is called “advantage A,” while if B wins the point it is “advantage B.” If the game is at advantage A and player A wins the next point, then player A wins the game. If player B wins the point at advantage A, the game is back at deuce.

a.    Suppose the probability of player A winning any point is p. Model the progress of a tennis game starting at deuce using a Markov chain with the following five states.

1. 1. deuce
2. 2. advantage A
3. 3. advantage B
4. 4. A wins the game
5. 5. B wins the game

Find the transition matrix for this Markov chain.

b.    Let p = .6. Find the probability that the game will be at “advantage B” after three points starting at deduce.