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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 jailor 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 jailor's reasoning?

[solution]

Let Ci be the ith prisoner will be executed. Ji be that jailor tells that the ith will be free. Suppose the 1st prisoner is asking. Then

P(Ci) = 1/3

P(J3|C2) = 1

P(J2|C3) = 1

P(J2|C1) = P(J3|C1) = 1/2

Compute [tex]P(C1|J2)[/tex] = [tex]\frac{P(C1 & J2)}{P(J2)}[/tex] = [tex]\frac{P(J2|C1)P(C1)}{(P(J2|C1)P(C1) + P(J2|C3)P(C3))}[/tex] = [tex]1/3[/tex]

By the same token P(C1|J3) = 1/3 => Jailor's reasoning is wrong.

I don't understand how the probabilities in red were computed.

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# Something like the Monty Hall Dilemma

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