When a tennis player serves he gets two chances to serve in

When a tennis player serves, he gets two chances to serve in bounds. If he fails to do so twice, he loses the point. If he attempts to serve an ace, he serves in bounds with probability 3/8. If he serves a lob, her serves in bounds with probability 7/8. If he serves an ace in bounds, he wins the point with probability 2/3. With an inbounds lob, he wins the point with probability 1/3. If the cost is +1 for each point lost and -1 for each point won, the problem is to determine the optimal serving strategy to minimize the (long-run) expected average cost per point. (Hint: Let state 0 denote the point over, two serves to go on next point; and let 1 denote one serve left)

Solution

State 0 State 1 State 2

over next point one serve left

3/8