Monte Hall. This is a famous (and surprisingly difficult) problem based on an old U.S.

television game show “Let’s Make a Deal hosted by Monte Hall”. A standard part of the show ran as follows: A contestant was asked to select from one of three identical doors: A, B, and C. Behind one of the three doors there was a prize. If the contestant selected the correct door they would receive the prize.

The contestant picked one door (say A) but it is not immediately opened. To increase the drama the host opened one of the two remaining doors (say door B) revealing that that door does not have the prize. The host then made the offer: “You have the option to switch your choice” (e.g. to switch to door C). You can imagine that the contestant may have made one of reasonings (a)-

(c) below. Comment on each of these three reasonings. Are they correct?

(a) “When I selected door A the probability that it has the prize was 1/3. No information was revealed.

So the probability that Door A has the prize remains 1/3.”

(b) “The original probability was 1/3 on each door. Now that door B is eliminated, doors A and C each have each probability of 1/2. It does not matter if I stay with A or switch to C.”

(c) “The host inadvertently revealed information. If door C had the prize, he was forced to open door B. If door B had the prize he would have been forced to open door C. Thus it is quite likely that door C has the prize.”

(d) Assume a prior probability for each door of 1/3. Calculate the posterior probability that door A and door C have the prize. What choice do you recommend for the contestant?