Suppose a series of games is played between two teams and that in any individual game, one of the teams wins and the other team loses. Let the probability that team A wins an individual game be p and hence the probability that team B wins is q = 1 ? p. Assume that the outcomes of individual games are independent. The series continues until one team wins N games in which case they are declared the overall winner.
Hint: See example 4j on page 81 in the 9th edition of Ross, the problem of points.
(a) If N = 2, what is the probability that A is the overall winner? (Your answer will depend on p).
(b) If N = 2 and X is the number of games needed to determine the overall winner, what are the possible values of X? What are the expected value, variance, and standard deviation of X? (Your answer should be a function of p and no other variables).
(c) What value of p maximizes the expected value of X when N = 2?