(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, 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.

Solution: (a.) Using the law of total probability, we can find the unconditional probability of winning in the same way as in class and we get 2/3. (b.) A tree method or Bayes' Rule with law of total probability will give both give 1/(1+p). (c.) The structure of the problem is the same as part b (except for the condition that p is greater than or equal to 1/2, which was no needed above). By the previous part, we get 1/(2-p).

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