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If MH's guess is random, the problem doesn't work out like this. It depends on the fact that his guess is not random. If his guess *IS* random, then the probability is what everyone else thinks it is.



If MH shows a door with the prize behind it then the contestant would pick that door! What makes the problem intersting is the counter-intuitive nature of the result favored by most mathematicians. The facile case in which MH reveals the prize thereby giving it away to the contestant (who after all may pick any door they wish) has no interest. Why do people bring it up?