Consider the Monty Hall problem, except that when Monty has a choice between opening doors 2 and 3, he opens door 2 with probability p, where 1/2 ≤ p ≤ 1. To recap: there are three doors, behind one of which there is a car (which you want), and behind the other two 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, which for concreteness we assume is door 1. Monty Hall then opens a door to reveal a goat, and offers you the option of switching. Assume that Monty Hall knows which door has the car, will always open a goat door and offer the option of switching, and as above assume that if Monty Hall has a choice between opening door 2 and door 3, he chooses door 2 with probability p. (a) Find the unconditional probability that the strategy of always switching succeeds (uncondi- tional in the sense that we do not condition on which of doors 2 or 3 Monty opens). (b) Find the probability that the strategy of always switching succeeds, given that Monty opens door 2. (c) Find the probability that the strategy of always switching succeeds, given that Monty opens door 3.
Consider the Monty Hall problem, except that when Monty has a choice between opening doors
2 and 3, he opens door 2 with
1/2 ≤ p ≤ 1. To recap: there are three doors,
behind one of which there is a car (which you want), and behind the other two 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, which for concreteness we assume is door 1. Monty Hall then opens a door to reveal
a goat, and offers you the option of switching. Assume that Monty Hall knows which door has the
car, will always open a goat door and offer the option of switching, and as above assume that if
Monty Hall has a choice between opening door 2 and door 3, he chooses door 2 with probability p.
(a) Find the unconditional probability that the strategy of always switching succeeds (uncondi-
tional in the sense that we do not condition on which of doors 2 or 3 Monty opens).
(b) Find the probability that the strategy of always switching succeeds, given that Monty opens
door 2.
(c) Find the probability that the strategy of always switching succeeds, given that Monty opens
door 3.
(d) Given that Monty opens door 2, is there a value of p such that switching has a lower success
probability than staying with the initial choice?
(e) Given that Monty opens door 3, is there a value of p such that switching has a lower success
probability than staying with the initial choice?
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