Quite a few years ago, a popular show called Let’s Make a Deal appeared on network television. Contestants were selected from the audience. Each contestant would bring some silly item that he or she would trade for a cash prize or a prize behind one of three doors.
Suppose that you have been selected as a contestant on the show. You are given a choice of three doors. Behind one door is a new sports car. Behind the other doors are a pig and a chicken—booby prizes to be sure! Let’s suppose that you pick door number one. Before opening that door, the host, who knows what is behind each door, opens door two to show you the chicken. He then asks you, “Would you be willing to trade door one for door three?” What should you do?
Required Tasks:
1. Given that there are three doors, one of which hides a sports car, calculate the probability that your initial choice is the door that hides the sports car. What is the probability that you have not selected the correct door?
2. Given that the host knows where the sports car is, and has opened door 2, which revealed a booby prize, does this affect the probability that your initial choice is the correct one?
3. Given that there are now only two doors remaining and that the sports car is behind one of them, is it to your advantage to switch your choice to door 3?