Jay and Maurice are playing a tennis match. In one particular game, they have reached deuce, which means each player has won three points. To finish the game, one of the two players must get two points ahead of the other. For example, Jay will win if he wins the next two points (JJ), or if Maurice wins the next point and Jay the three points after that (MJJJ), or if the result of the next six points is JMMJJJ, and so on.
a. Suppose that the probability of Jay winning a point is .6 and outcomes of successive points are independent of one another. What is the probability that Jay wins the game? [In the law of total probability, let A1 = Jay wins each of the next two points, A2 = Maurice wins each of the next two points, and A3 = each player wins one of the next two points. Also let p = P(Jay wins the game). How does p compare to P(Jay wins the game | A3)?]
b. If Jay wins the game, what is the probability that he needed only two points to do so?