I believe that you are stating the problem correctly, but then adding a big assumption before analyzing it. You assume that that, whether or not you have the right choice, the host will always put you in the current situation - you've picked a door, and another has been opened showing a goat. With that assumption then you should switch, and will have 2/3 odds of winning if you do.

But that is a big assumption. In analyzing probabilities you cannot just work with what has happened, you have to work with what could have happened instead. A probability problem is never fully specified until it includes both knowledge of what did happen and what might have happened.

In particular in this case there are models of the host's possible behaviour in which you would be an idiot to switch. (The host is trying to keep you from winning the car.) There are models of the host's behaviour in which it is a good idea to switch. (The host wants to draw the game out.) And the problem statement does not provide enough information to unambiguously decide which model of the host's behaviour is correct. (Saying that the host has enough knowledge to always draw the game out is not saying that the host will choose to do that.)

Therefore the problem is not fully specified. To come up with an answer you have to add an assumption of some sort.

If you want to make it truly unambiguous, you can state the problem like this, Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. The game works like this; you pick a door and the host, who knows what's behind the doors, will open another door and show you a goat. Then the host asks whether you want to switch your choice. You decide, then the host opens the other doors and you get the car if you've chosen it. When you play this game, is it to your advantage to switch your choice of doors?

The wording change is subtle but important. With this wording it is clear that, no matter what, the first time the host opens a door there will always be a goat. Not only have you been told what actually happened in the game, but you've also been told what would've happened no matter what your initial choice was.

In reply to Re^7: Marilyn Vos Savant's Monty Hall problem by tilly
in thread Marilyn Vos Savant's Monty Hall problem by mutated

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