A debate recently erupted about the optimal strategy for playing a game on the TV show called "Let's Make a Deal: In one of the games on this show, the contestant would be given the choice of prizes behind three closed doors. A valuable prize was behind one door and worthless prizes were behind the other two doors. After the contestant selected a door, the host would open one of the two remaining doors to reveal one of the worthless prizes. Then, before opening the selected door, the host would give the contestant the opportunity to switch his or her selection to the other door that had not been opened. The question is, should the contestant switch?
a. Suppose a contestant is allowed to play this game 500 times, always picks door number 1, and never switches when given the option. If the valuable prize is equally likely to be behind each door at the beginning of each play, how many times would the contestant win the valuable prize? Use simulation to answer this question.
b. Now suppose the contestant is allowed to play this game another 500 times. This time, the player always selects door number 1 initially and switches when given the option. Using simulation, how many times would the contestant win the valuable prize?
c. If you were a contestant on this show, what would you do if given the option of switching doors?