Three prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information, since he already knows that at least one will go free. The jailer refuses to answer this question, pointing out that if A knew which of his fellows were to be set free, then his own probability of being executed would rise from 1/3 to 1/2 , since he would then be one of two prisoners. What do you think of the jailer’s reasoning?
The problem presented here is a classic puzzle in probability theory, known as the "Three Prisoners Problem." The puzzle is interesting because it challenges our intuition about conditional probability and demonstrates the importance of careful reasoning.
The jailer's reasoning is correct. At the beginning of the problem, each prisoner has an equal chance of being executed, which is 1/3. However, if the jailer were to reveal which of the other two prisoners would be set free, then prisoner A's probability of being executed would increase to 1/2. This is because if prisoner A knew that one of the other prisoners was guaranteed to be set free, then the only two possible outcomes would be either prisoner A is executed, or he is set free along with the other prisoner. In other words, his probability of being executed is now 1/2 instead of 1/3.
One way to think about this is to consider the two possible scenarios that could result from the jailer's decision. If prisoner A is told which of the other two prisoners is to be set free, then he knows that he is not in that group and therefore his probability of being executed has increased to 1/2. On the other hand, if prisoner A is not told which of the other two prisoners is to be set free, then he still has a 1/3 chance of being executed, but he also has a 2/3 chance of being set free with one of the other prisoners.
In conclusion, the jailer's reasoning is sound. If he were to reveal which of the other two prisoners would be set free, then prisoner A's probability of being executed would increase to 1/2. The solution to this puzzle demonstrates the importance of understanding conditional probability and carefully analyzing the different possible outcomes in a problem.
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