Use computer simulation to find the optimal value of the stopping level L and the maximal probability of candidate A winning.

Go back and take another look at Problem 2.29 from Chapter 2. For ease of notation, let us rename the numbers 5, 10,…, 100 on the wheel as 1, 2,…, 20. For any a = 1, 2,…, 20, let S(a) denote the probability of candidate A winning if candidate A stops after the first spin giving a score of a points and let C(a) denote the probability of candidate A winning if candidate A continues after the first spin giving a score of a points. Use conditional probabilities to find first an expression for S(a) and next an expression for C(a). Derive from these expressions the optimal stopping rule for candidate A and the maximal probability of candidate A winning. Repeat the calculations for the case where the numbers 1, 2,…, 100 are on the wheel rather than the numbers 1, 2,…, 20. Problem 2.29 Two candidates A and B remain in the finale of a television game show. At this point, each candidate must spin a wheel of fortune. The 20 numbers 5, 10,…, 95, 100 are listed on the wheel and when the wheel has stopped spinning, a pointer randomly stops on one of the numbers. Each candidate has a choice of spinning the wheel one or two times, whereby a second spin must immediately follow the first. The goal is to reach a total closest to but not exceeding 100 points. The winner is the candidate who gets the highest score. Should there be a tie, then the candidate to spin the wheel first is the winner. The candidate who spins second has the advantage of knowing what the score of the first candidate was. Lots are drawn to determine which player begins. Suppose that candidate A has to spin first. His/her strategy is to stop after the first spin if this spin gives a score larger than a certain level L and otherwise to continue for a second spin. Use computer simulation to find the optimal value of the stopping level L and the maximal probability of candidate A winning.

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