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| Answer: The right chair The odds are 1 in 2. Consider the following cases: Case 1: The intoxicated person enters the room last. This case is ruled out. Case 2: The intoxicated person is the 2nd to last to enter the room. His chances of picking the right chair are 1 in 2. Therefore, the chance that the last guy will be able to sit in the right chair are 1 in 2. Case 3: The intoxicated person is the 3rd to last to enter the room. His chances of picking the right chair are 1 in 3. If he does this, the last person will sit in the right chair. His chances of picking the wrong chair are 2 in 3. For the case where he picked the wrong chair, one of the last two people to enter the room will be the "displaced" person. In 1/2 of those cases, the displaced person will enter the room last, in which case he will get the intoxicated person's chair -- the wrong one. In the other 1/2 of the cases, he is in exactly the same boat as the intoxicated person was in case 2, where the "right" chair for him is now the intoxicated person's chair (so that the last person can get his own chair). The odds are therefore: 1/3*1 + 2/3*(1/2*0 + 1/2*1/2) = 1/3 + 1/6 = 1/2. Case 4: The intoxicated person is the 4th to last to enter the room. His chances of picking the right chair are 1 in 4. If he does this, the last person will sit in the right chair. His chances of picking the wrong chair are 3 in 4, in which case one of the others will be displaced. In 1/3 of those cases, the displaced person will enter the room last, in which case he will get the intoxicated person's chair -- the wrong one. In the other 2/3 of the cases, he is in exactly the same boat as the intoxicated person was in case 3. The odds are therefore: 1/4*1 + 3/4*(1/3*0 + 2/3*1/2) = 1/4 + 1/4 = 1/2. There's an inductive rule here... Case N: Suppose the odds at case N-1 are 1/2 where "N" means that the intoxicated person entered the room "Nth from last" (there are N empty chairs for the intoxicated person to choose from). Then the odds at N are: 1/N*1 + (N-1)/N * (1/(N-1)*0 + (N-2)/(N-1)*1/2) = 1/N + (N-2)/2N = 1/N + N/2N - 2/2N = 1/2 |
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