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After a person mis-evaluates, and says either that the chances after the goat is revealed go to 50-50, or that both of the doors retain their original 1/3 chance, or that, for whatever reason, he doesn't improve his chances by switching, present a modified version of the problem as follows:

The contestant picks a door. 1/3 chance per door of the car being behind it.

Monty says: I'm in an especially generous mood today, so I'm going to make you an offer: you can have both of the other two doors in exchange for your chosen door.

Should the contestant switch?

I think most people will recognize that the 2 doors between them have a 2/3 chance compared to the original 1/3 chance, so most people will say the contestant should switch.

So let's say in this version of the game that the contestant trades his 1 door for the 2 doors.

Monty then says: I'm going to open one of your 2 doors first. If neither door has the car, I don't care which one of them I open, but if one of them has the car, I'm going to open first the only one that doesn't. Either way, I'm going to open one of your doors that doesn't have the prize, whether it's the only one available for that or not, and I'm not going to open your originally chosen door right now.

The contestant isn't too worried about this, because he knows that only one of the doors could have the car anyway. He's anxious to see whether his 2/3 chance will pay off, but that's all that worries him in the game at that moment.

So Monty opens one of the contestant's 2 doors, and reveals a goat. No-one, including the contestant, is surprised, but a murmer comes from the audience.

Monty says: You traded in your 1/3 chance for what was then, at least then, a 2/3 chance. Now there are only 2 doors remaining. If you want to, I'll let you trade back for your original door."

The contestant becomes flustered, because for a moment, the audience murmers more loudly.

Should he switch back?

Monty then offers him 10% of the car's value to switch. If he's thinking his chances are "1 door 1/3" as at the start, or that they went up to 2/3 when he was allowed to switch for 2 doors, but now that there are only 2 unopened doors, his chances are now 50-50, the 10% cash should tip the scales for him, so he might think he should switch back.

Should he switch back?

I think most people will realize that, given what Monty said, the opening of a non-winning door of the 2 doors switched for won't diminish the 2/3 chance. The contestant knew when he switched for the 2 doors that at least one had to be a non-winner, because there's only 1 car in the game.

If a question respondent who thought in the original problem that there was no advantage to switching, thinks that in this version there's an advantage to not switching back, ask him to reconsider, after closer examination, whether in the original problem switching doors after the reveal is equivalent to not switching back after the reveal in the second version.

If he doesn't think so, ask him to imagine the game played both ways with the car behind the same door, and the same door originally chosen in each game. Why should the game-1 contestant not switch, if the game-2 contestant was right to switch before a door was opened and would be mistaken to switch back afterward?