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What if Monty only opens a door when he knows you've chosen the prize door?

Extension: what if Monty tosses a coin to decide whether to open a door, conditioned on you having chosen the prize door?

Specifying Monty's behaviour matters, I think.

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Monti's a priori knowledge absolutely matters and should have been stated in the article. If he had no knowledge beforehand about which door the prize was behind, then you would have to expand the probability table to include cases in which Monti opens one of the remaining two doors and accidentally reveals the prize, in which case the user just loses without getting the opportunity to switch or not switch. Monti knowing the door is hiding the prize and actively selecting the non-prize door versus just guessing and sometimes accidentally revealing the prize, are two different statistics problems.

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