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The wording of the classic problem is as follows:
'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. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?'
The answer is that statistically, it is always to your advantage to do so: contestants who switch have a 2/3 chance of winning the car, while contestants who stick to their choice have only a 1/3 chance. However, the words 'who knows what's behind the doors' are crucial, for as Wikipedia points out, 'If we assume the host opens a door at random, when given a choice, then which door the host opens gives us no information at all as to whether or not the car is behind door 1.' In that case, the player's odds when changing doors are 1/2:1/2, not 2/3:1/3.
In the given puzzle, the text is silent as to whether the host knows what's behind the doors. If he does know, then the expected value of changing or sticking is exaclty the same: (4000*2/3 + 1000*1/3) = (2000*2/3 + 5000*1/3). If he does not know, then it is better to stick with the original box, since (1/2*2000+1/2*5000) exceeds (1/2*1000+1/2*4000). Note here that we need not know what the probability is that the host did or didn't know what was behind the doors; sticking with the first box at worst gives an equal pay-off, and can give a higher payoff, so it is always better to stick.
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