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I imagine the following setup.

There is a contestant who has to pick one of three doors. How many prizes will be hidden is determined at random in the following way. Monty will toss a fair coin and if it comes up heads 2 prizes will be hidden behind the 3 doors, whilst if it comes up tails only 1 prize will be hidden. The contestant is informed about the rules, but Monty will not reveal any information to the contestant regarding the outcome of the toss.

The contestant now picks a door, and the game only continues if there is a prize behind it. However in order to win the prize the contestant will have to tell Monty what he thinks the result of the toss was ! If he s wrong he will not win anything.

The Question is what should the contestant answer, and does it matter what is answer is ? (naturally it is assumed he wants to win the prize)

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Not sure if this is right, but intuitively it seems to be more logical to guess heads. – itdoesntwork Jan 29 '12 at 18:34
up vote 1 down vote accepted

At the point at which the contestant has chosen a door, there are $18$ equally likely outcomes: two ways for the coin to come up, in each case three ways for the prizes to be distributed, and three ways for the contestant to guess. In six of them the coin came up heads and the contestant picked a door with a prize behind it, and in three of them the coin came up tails and the contestant picked the door with the prize behind it. (I am assuming that when the coin comes up heads, the two prizes are put behind different doors.) Given that the contestant picks a potentially winning door, the odds are $2:1$ that the coin came up heads: if he guesses heads, his probability of getting the prize is $2/3$, while if he guesses tails, it is only $1/3$. Assuming that he guesses heads, his overall probability of winning a prize is $(1/2)(2/3)=1/3$, since his probability of choosing a potentially winning door is $1/2$.

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absolutely, thanks for pointing out the ambiguity, but prizes will be placed behind different doors ! – Beltrame Jan 29 '12 at 18:42
Here is my question to you, isn t this problem essentially the same as the sleeping beauty problem in the sense that once we condition on the state that we find ourselves in we have to readjust our conditional estimate of the outcome of the coin toss, even tough it s a fair coin ? – Beltrame Jan 29 '12 at 19:45
@PeeJay: I’ve not entirely decided what I think about the sleeping beauty question, though my initial preference is for the ‘thirder’ position. – Brian M. Scott Jan 29 '12 at 19:56

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