The World Series recently concluded. One could try to model it by saying team A wins each game with probability p, team B wins with probability q, and the games are independent of each other. For baseball this is not a good model, because the win probability varies from game to game according to who the starting pitchers are, home field advantage, and possibly other things. The assumption of independence is also questionable because of fatigue or injuries.There are however many other games, for example table tennis, for which this is a good model. Suppose A and B play a best of 7 series, where the series ends when one player has won 4 games. Suppose A wins each gamewith probability P, B wins each game with probability q (where p + q = 1),and the games are independent.

7. Find the probability that A Wins the series 43 (A must Win game 7 and three of the rst six games). Find the probability that B wins 43. What is the probability f7(p, q) that the series ends after 7 games? 8. Again it is not interesting to ask for the minimum of f 7(1)) (0- What value or values of p maximize f7 (p, q), and What is the corresponding value or values of q? (The calculation may again be easier if you express fp, q) in terms of the variable u 2 pg 2 p(1 19).)