Question

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 \(\frac{1}{3}\) to \(\frac{1}{2}\), since he would then be one of two prisoners. What do you think of the jailer's reasoning?

Short answer

In conclusion, the jailer's reasoning is partially correct, in that knowing which prisoner would go free does indeed change prisoner A's probability of being executed in the context of just A and the remaining prisoner. However, this line of reasoning overlooks the fact that the overall probability of prisoner A being executed, without any additional information, is still \(\frac{1}{3}\), as the increase in conditional probabilities does not necessarily imply a change in the initial probabilities.

Step-by-step solution

Understand the initial probabilities

Initially, there are three prisoners, and one will be executed. Therefore, the probability for each prisoner to be executed is equal to \(\frac{1}{3}\).

Understand the request of prisoner A

Prisoner A wants to know which of his fellow prisoners, B or C, will be set free. Keeping in mind that one of B or C will definitely be set free (because only one prisoner is to be executed), we can determine the conditional probabilities.

Evaluate the probabilities if prisoner A knows more information

If the jailer tells prisoner A that prisoner B will be set free, then all the probability goes to either prisoner A or prisoner C being executed. In this case, the probability of prisoner A being executed is indeed \(\frac{1}{2}\). However, if the jailer tells prisoner A that prisoner C will be set free, then the same logic would imply that the probability of prisoner A being executed is again \(\frac{1}{2}\).

Acknowledge the key point in the jailer's reasoning

The jailer's refusal to answer is based on the idea that if prisoner A knew which of his fellow prisoners would be set free, his own probability of being executed would increase. This is technically correct, as given the information about who is set free, indeed the probability of being executed for 'A' would rise to \(\frac{1}{2}\). However, the jailer's logic does not consider that both these conditional events already have a probability of occuring and the fact that either one of the remaining prisoners will be set free makes no impact on Prisoner A's overall probability of being executed.

Evaluate the jailer's overall reasoning

In conclusion, the jailer's reasoning is partially correct, in that knowing which prisoner would go free does indeed change prisoner A's probability of being executed in the context of just A and the remaining prisoner. However, this line of reasoning overlooks the fact that the overall probability of prisoner A being executed, without any additional information, is still \(\frac{1}{3}\), as the increase in conditional probabilities does not necessarily imply a change in the initial probabilities.