Fall 2011 Homework 3: Question 1
(a) Consider the following 7-door version of the Monty Hall problem. There are 7 doors, behind one of which there is a car (which you want), and behind the rest of which there are goats (which you don't want). Initially, all possibilities are equally likely for where the car is. You choose a door. Monty Hall then opens 3 goat doors, and offers you the option of switching to any of the remaining 3 doors. Assume that Monty Hall knows which door has the car, will always open 3 goat doors and offer the option of switching, and that Monty chooses with equal probabilities from all his choices of which goat doors to open. Should you switch? What is your probability of success if you switch to one of the remaining 3 doors?
(b) Generalize the above to a Monty Hall problem where there are $n\geq 3$ doors, of which Monty opens m goat doors, with $1\leq m\leq n-2$.
Solution: (a) Assume the doors are labeled such that you choose Door 1 (to simplify notation), and suppose first that you follow the "stick to your original choice" strategy. Let S be the event of success in getting the car, and let Cj be the event that the car is behind Door j. Conditioning on which door has the car, we have P(S) = P(S|C1)P(C1) + · · · + P(S|C7)P(C7) = P(C1) = 1/7. Let Mijk (for all i, j, k between 2 and 7 inclusive) be the event that Monty opens Doors i, j, k. P(S|Mijk) = P(S) = 1/7 by symmetry. Thus, the conditional probability that the car is behind 1 of the remaining 3 doors is 6/7, which gives 2/7 for each. So you should switch, thus making your probability of success 2/7 rather than 1/7. (b) By the same reasoning, the probability of success for "stick to your original choice" is 1/n, both unconditionally and conditionally. Each of the n − m − 1 remaining doors has conditional probability n−1/(n−m−1)n of having the car. This value is greater than 1/n, so you should switch, thus obtaining probability n−1/(n−m−1)n of success (both conditionally and unconditionally).
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."