TITLE: A variant of the Monty Hall problem
QUESTION [44 upvotes]: Everybody knows the famous Monty Hall problem; way too much ink has been spilled over it already. Let's take it as a given and consider the following variant of the problem that I thought up this morning.
Suppose Monty has three apples. Two of them have worms in them, and one doesn't. (For the purposes of this problem, let's assume that finding a worm in your apple is an undesirable outcome). He gives three "contestants" one apple each, then he picks one that he knows has a worm in his apple and instructs him to bite into it. The poor contestant does so, finds (half of) a worm in it, and runs off-stage in disgust.
Now consider the situations of the two remaining contestants. Each one has a classical Monty Hall problem facing him. From player A's perspective, one "door" has been "opened" and revealed to have a "goat"; using the same logic as before, he should choose to switch apples with player B.
The paradox is that player B can use the same logic to conclude that he should switch apples with player A. Therefore, each of the two remaining contestants agree that they should switch apples, and they'll both be better off! Of course, this can't be the case. Exactly one of them gets a worm no matter what.
Where is the flaw in the logic? Where does the analogy between this variant of the problem and the classical version break down?

REPLY [2 votes]: There is a way to get the Monty Hall problem back. But it requires to wiggle a bit on the assumptions. I.e. the problem does not specify how Monty is choosing the candidate that has to bite into the apple first. He could of course choose this candidate at random among the two candidates with the worm.
But let's assume that in order to boost ratings, Monty always spares the most likeable person from biting into the apple first. Assume further that you are actually the most likeable person in the show. So you know that you are not picked first. Then after one candidate is kicked off the show, you do actually face the original Monty Hall problem:
I.e. you initially have a bad apple with probability 2/3, but by switching you can decrease the chance of a bad apple to 1/3.
The fine difference between the situation here and the situation analyzed in the other answers is that here you know (assuming that you are the most likeable person and Monty is the rating-boosting selection strategie) that you don't get kicked off in the first round.