Exercises 35–40 concern the two Markov chain models for scoring volleyball games described in Section 10.1, Exercise 36. Suppose that teams A and B are playing a 15-point volleyball game that is tied 15-15 with team A serving. Suppose that the probability p that team A wins any rally for which it serves is p = :7, and the probability q that team B wins any rally for which it serves is q = :6.

Since p = .7 and q = .6, it seems that team A is the dominant team. Does it really matter which scoring system is chosen? Should the manager of each team have a preference?

Data From Section 10.1 Exercise 36

Volleyball uses two different scoring systems in which a team must win by at least two points. In both systems, a rally begins with a serve by one of the teams and ends when the ball goes out of play or touches the floor or a player commits a fault. The team that wins the rally gets to serve for the next rally. Games are played to 15, 25, or 30 points.
a. In rally point scoring, the team that wins a rally is awarded a point no matter which team served for the rally. Assume that team A has probability p of winning a rally for which it serves, and that team B has probability q of winning a rally for which it serves. Model the progress of a volleyball game using a Markov chain with the following six states.
1 tied – A serving
2 tied – B serving
3 A ahead by 1 point – A serving
4 B ahead by 1 point – B serving
5 A wins the game
6 B wins the game
Find the transition matrix for this Markov chain.
b. Suppose that team A and team B are tied 15–15 in a 15- point game and that team A is serving. Let p = q = .6.
Find the probability that the game will not be finished after three rallies.
c. In side out scoring, the team that wins a rally is awarded a point only when it served for the rally. Assume that team A has probability p of winning a rally for which it serves, and that team B has probability q of winning a rally for which it serves. Model the progress of a volleyball game using a Markov chain with the following eight states.
1 Tied – A serving
2 Tied – B serving
3 A ahead by 1 point – A serving
4 A ahead by 1 point – B serving
5 B ahead by 1 point – A serving
6 B ahead by 1 point – B serving
7 A wins the game
8 B wins the game
Find the transition matrix for this Markov chain.
d. Suppose that team A and team B are tied 15 – 15 in a 15- point game and that team A is serving. Let p = q = .6. Find the probability that the game will not be finished after three rallies.