Three prisoners A,B, and C have been sentenced to die tomorrow. The king has cho

Question

Three prisoners A,B, and C have been sentenced to die tomorrow. The king has chosen one the three uniformly at random to be pardoned tomorrow, while the two unlucky ones head for the gallows. The guard already knows who is to be pardoned. Prisoner A begs the guard to name someone other than A himself who will be executed. He cajoles, "Even if you tell me which one of B and C will be executed, I will not have gained any knowledge because I know already that at least one of them will die tomorrow." The guard is persuaded and reveals that B is to die tomorrow.

a. After receiving this information, does A still have probability 1/3 of being pardoned?

b. Prisoner A whispers his new information to prisoner C. Prisoner C learned conditional probability before turning to a life of crime and is now hopeful. What is his new probability of being pardoned?

Explanation / Answer

a) Since it is known for sure that B is going to be executed, either of A or C will be pardoned tomorrow. Hence, Probabiltiy of pardon=1/2

b) New probbility of being pardone = P( C is Pardoned/ B is not pardoned) = P( C is pardoned and B is not pardoned)/P( B is not pardoned) = 1/3*2/3 /[2/3]=1/3

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