The Ruritanian authorities decided to pardon and release one out of three remaining inmates, $A, B$ and $C$, 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 $A$ and offers to name, in exchange for a fee, another inmate (not $A)$ who is doomed to stay. He says: "This reduces your chances to remain here from $2 / 3$ to $1 / 2$ : will it make you feel better?" $A$ hesitates but then accepts the offer; the guard names $B$.

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

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

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

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

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

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