"But we must remember, Monty can always open a door with a non-prize behind it. We don't know if he had a choice in which door to open."

Very astute observation - and this was one of the reasons that MVS had such trouble. People did not assume this game was exactly like the game show. In the game show, Monty Hall has perfect knowledge and we KNOW that he never turns over the grand prize - only the goat!

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.