Question: Consider the Monty Hall problem, except that Monty enjoys opening door 2 more than he enjoys opening door 3, and if he has a choice between opening these two doors, he opens door 2 with probability p, where ½ ≤ 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 o↵ers you the option of switching. Assume that Monty Hall knows which door has the car, will always open a goat door and o↵er 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 (with ½ ≤ p ≤ 1)
(a) Find the unconditional probability that the strategy of always switching succeeds (unconditional 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.