Two different scoring systems exist in volleyball 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. We can model the end of a volleyball game starting from a tied score 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 team B is serving. Let p = q = 0.55. Find the probability that the game will not be finished after four rallies.
c) In side out scoring, the team that wins a rally is awarded a point only if it served for that 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. We can model the end of a volleyball game starting from a tied score 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 25-25 in a 25-point game and team B is serving. Let p = q = 0.7. Find the probability that the game will not be finished after three rallies.