Probability | Part IA, 2001

The Ruritanian authorities decided to pardon and release one out of three remaining inmates, A,BA, B and CC, kept in strict isolation in the notorious Alkazaf prison. The inmates know this, but can't guess who among them is the lucky one; the waiting is agonising. A sympathetic, but corrupted, prison guard approaches AA and offers to name, in exchange for a fee, another inmate (not A)A) who is doomed to stay. He says: "This reduces your chances to remain here from 2/32 / 3 to 1/21 / 2 : will it make you feel better?" AA hesitates but then accepts the offer; the guard names BB.

Assume that indeed BB will not be released. Determine the conditional probability

P(A remains B named )=P(A&B remain )P(B named )P(A \text { remains } \mid B \text { named })=\frac{P(A \& B \text { remain })}{P(B \text { named })}

and thus check the guard's claim, in three cases:

a) when the guard is completely unbiased (i.e., names any of BB and CC with probability 1/21 / 2 if the pair B,CB, C is to remain jailed),

b) if he hates BB and would certainly name him if BB is to remain jailed,

c) if he hates CC and would certainly name him if CC is to remain jailed.

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