Three prisoners are informed by their jailer that one of them has been chosen at random to be executed ...?

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 one of his fellow prisoners that will be set free, claiming that there would be no harm in divulging this information because he already knows that at least one of the two will go free. The jailer refuses to answer this question, pointing out that if A knew which of his fellow prisoners were to be set free, then his own probability of being executed would rise from 1/3 to 1/2 because he would then be one of two prisoners. What do you think of the jailer’s reasoning?

Answer 1

This is an interesting variation of the Monty Hall problem. Let's imagine that some in the public are making wagers on who will be executed. We'll call the prisoners A, B, C. Before there was any talk between A and the jailer, the odds of A having already been selected for execution is 1/3, because that's how the selection was conducted. We have 3 possible cases, the selected man being either A, B, or C. Let's say that the jailer agrees to release either B or C, that have not been selected for execution, leaving 2 prisoners. The betting public is very likely to feel that now the odds of A being executed is 1/2, since there are only 2 prisoners left. But in fact, there are still 3 possible cases, which is that prisoner A is the selected one, and jailer the frees either B or C, or B is the selected on, and jailer the frees C, or C is the selected one, and the jailer frees B, so that the odds of A being the selected one is still 1/3, while the odds of the remaining B or C being the selected one is now 2/3. The jailer's explanation to A why he should not release a prisoner should be that it's because then the other prisoner besides A would find his own odds of being the selected one doubled, and so an act of kindness would be balanced by an act of unkindness.

Answer 2

the one to be executed will be chosen at RANDOM out of the three... so even the jailer do not know which one of the three is to be executed... therefore the probability remains 1/3.

Answer 3

Prisoner A already knows that he will be executed or set free.

His chances are 50-50. If the guard had told him which of the other prisoners was being freed, Prisoner A would still know that his chances are 50-50. It wouldn' have made any difference if the guard had told told him who was being freed.

Answer 4

It stays at 1/3.

Answer 5

The jailer's reasoning makes no sense.