A forerunner to current game shows likeWho Wants to Be a MillionaireandDeal or No DealwasLet's Make a Deal. Contestants onLet's Make a Dealwould come dressed in weird costumes. They would try to gain the attention of host Monty Hall. The ultimate goal of each of the contestants was to be selected for the final deal of the day. In this deal some lucky contestant was faced with the choice of selecting one of three doors. Two of the doors contained prizes of minimal value like a box of cornflakes or a goat and one of the doors contained the grand prize of the day - perhaps a red Corvette or a cruise or a vacation with all expenses paid to some exotic location.

• What is the probability you will initially choose a door with the new car behind it? Please elaborate on how you calculated this probability. What is the probability that you will initially choose a door with a goat behind it? Please elaborate on how you calculated this probability.

• Next, the host opens one of the doors you didn't select and shows you that there is a goat behind it. He then offers you the choice of sticking with your first selection or switching to the other unopened door. Will the host's actions increase your chances of winning? If so, by how much? What will be your probability of winning the car given the host has opened agoat door? Generally speaking, since agoat dooris open, do you think you should stick with your initial choice or that you should switch to a different door; or, do you think there will be no difference in winning by sticking or winning by switching?
• run a simulation of this game. Begin by selecting one person to be the host; the other will be the contestant. The host shouldrandomlyselect one of the doors (perhaps by drawing a number out of a hat or by some other means of generating a random number between 1 and 3), and the selection must not be revealed to the contestant. The contestant should now select one of the three doors. After the contestant reveals which door has been chosen, the host should select agoat doorand inform the contestant that this particular door has a goat behind it. The host should now offer the contestant a choice of sticking with the original door or switching to the other door that has not been revealed. After the contestant announces his or her decision to stick or switch, the host should announce that the contestant has or has not won the Corvette. Record the result along with the contestant's decision to stick or switch. Repeat the game 20 times with the contestant sticking 10 times and switching 10 times (in whatever order you choose). Record the results of your experiment by completing Table Ia and Table Ib.
• What proportion of the games won was won by sticking? To calculate this probability, count the number of wins you recorded. This will become the denominator of the probability fraction. Now count the number of times you won by sticking. This will become the numerator of the probability proportion. What proportion of the games won was won by switching? You have already counted the number of wins. This number is still the denominator of the probability fraction. Now count the number of times you won by switching. This will become the numerator of the probability fraction.
• what interpretation can you offer about which is the better strategy - to stick or to switch?