2.15 Project: Strategy to Win

Consider Example 2.15 of the textbook (Example 2.14 of the slides). Suppose that the player chooses door A and the host opens door B.

(a) Conduct a computer experiment to verify the solution that the probability of winning by switching is 2/3. Report your design and computer code of the experiment and average results from 10, 000 trials of the experiment.

(b) After the host opens door B, the player flips a fair coin and gets the tail, and this entails him to switch to door C. What is his probability of winning?

(c) Determine the winning probability of the following strategy: after the host opens door B, the player flips a fair coin, and he will stick to door A if he gets the head, otherwise he will switch to door C.

(d) Suppose that without knowing the player's choice (door A) the host opens at random one of the doors without the car. What is the probability for the player to win by never switching the door?

(e) Suppose that without knowing the player's choice (door A) the host opens at random one of the doors without the car. What is the best strategy for the player to win and what is its winning probability? If the host opens by chance door A, what is the probability for the player's best strategy to win? If the host opens by chance door B, what is the probability for the player's best strategy to win?

(f) Conduct a computer experiment to verify your solutions to parts (b)-(e) above. Report your design and computer code of the experiment and average results from 10, 000 trials of the experiment.

Consider now the general n-door problem in Example 2.15. Suppose that the player chooses

door A and the host opens all remaining doors except say, B.

(g) Whatistheprobabilityfortheplayertowinbyalwaysswitchingtotheonlyremainingdoor?

(h) SupposethatafterthehostopensallremainingdoorsexceptB,theplayerflipsafaircoin,gets

the tail, and this entails him to switch to door B. What is his probability of winning?

(i) Suppose that without knowing the player's choice (say, A) the host opens at random (n 2) doors without the car, which turns out to be different from the player's choice. What is the probability for the player to win by always switching to the only remaining door (say, B)?

What is the probability for the player to win by never switching the door?

(j) Conductacomputerexperimenttoverifyyoursolutionstothen-doorproblemforanon-trivial special case (such as a 5-door problem) for cases (g)-(j). Report your design and computer

code of the experiment and average results from 10, 000 trials of the experiment.