Assume that the probability of winning a point on serve or return is treated as constant throughout the match. Further suppose that x is the probability that the better player in a match wins a set.

(a) The probability P₃ that the better player wins a best-of-three match is P₃ (x) = x [1 + 2(1 − x)]. Suppose the probability that the better player wins a set is 0.6. What is the probability that this player wins a best-of-three match?

(b) The probability P₅ that the better player wins a best-of-five match is P₅ (x) = x³ [1 + 3(1 − x) + 6(1 − x)²]. Suppose the probability that the better player wins a set is 0.6. What is the probability that this player wins a best-of-five match?

(c) The difference between the probability of winning and losing is known as the win advantage. For a best-of-n match, the win advantage is P_n − (1 − P_n) = 2P_n − 1. The edge, E, is defined as the difference in win advantage between a best-of-five and best-of-three match. That is, E = (2P₅ − 1) − (2P₃ − 1) = 2(P₅ − P₃). Edge as a function of win probability of a set x is E(x) = 2x² { x[ 1 + 3(1 − x) + 6(1 − x)²] − (1 + 2(1 − x))} Graph E = E(x) for 0.5 ≤ x ≤ 1.

(d) Find the probability of winning a set x that maximizes the edge. What is the maximum edge?

(e) Explain the meaning of E(0.5).

(f) Explain the meaning of E(1).