The World Series recently concluded. One could try to model it by say- ing 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 ac- cording 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 game with probability p, B wins each game with probability q (where p + q = 1), and the games are independent.3. Find the probability that A wins the series 4-1 (A must win game 5 and three of the first four games). Find the probability that B wins 4-1. What is the probability f5(p, q) that the series ends after 5 games? 4. It is clear that the minimum of fs(p, q) occurs when p = 1 or q = 1, since the series must end in 4 games in those cases. What value or values of p maximize fs(p, q), and what is the corresponding value or values of q? Hint: The calculation may be easier if you express f5(p, q) in terms of the variable u = pq = p(1 - p), noting that du = 1 - 2p. dp