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A certain prison has 100 prisoners. The warden offers the following deal: Each prisoner has a unique number from 1 to 100. In one room, 100 boxes stand in a row. Into each box is placed a scrap of paper containing a number from 1 to 100. Each number appears exactly once, and the order is random. The prisoners will be allowed to enter the room one at a time, and open boxes one at a time, up to 50 boxes. They may only look in the box to see what number is there, they may not move the scraps of paper or the boxes. All boxes will be closed after the prisoner leaves the room, before the next prisoner is brought in. If any prisoner fails to open the box containing their own number, all the prisoners will be put to death. If every prisoner finds their own number, then all prisoners will go free. Once the process begins, prisoners will not be allowed to communicate with each other in any fashion. Clearly if each prisoner opens 50 boxes at random, the probability that everyone will go free is (1/2)^100, which is not very good. By what strategy can the prisoners improve their chances?
3 Answers
- JazSincLv 76 years ago
Each prisoner has a 50-50 chance of opening the box with his own number.
For every prisoner to pick the box with his own number, that'd be like flipping a coin 100 times and have it come up heads each time.
So, if the deal is taken, the only scenario is the one in which all the prisoners are put to death.
The prisoners stay alive by not accepting the warden's deal.
- Anonymous6 years ago
Well, they can improve their odds slightly by communicating which boxes each prisoner will open beforehand. The only additional information the prisoners will get after the process begins will be knowing whether or not each prisoner is successful after going through the process. So, if everyone agrees that prisoner #1 will open the first 50 boxes on the left, then, if he is successful, all ensuing prisoners will know that box #1 is on the left half. Which means that when prisoner #2 goes in he has a 50/99 chance if he opens all the boxes on the right, but only a 49/99 chance if he opens boxes on the left.
It's not MUCH of an improvement, but it's the only thing I can think of.
