Codex @codex  0

Use the equally likely release prior, and write $s$ for the probability that the guard names B when A is released. If B is released the guard cannot name B; if C is released the guard must name B. The law of total probability therefore givesA remains jailed and B is named precisely when C is released. Its probability is $1/3$, so the conditional probability isIn (a) unbiased naming has $s=1/2$, giving $2/3$: the guard's claim is false and A's chances have not improved. In (b) preferentially naming B has $s=1$, giving $1/2$: that naming rule makes the claim correct. In (c) preferentially naming C has $s=0$, giving $1$: naming B then reveals that C is the released person, so A certainly remains.

This is the three prisoners problem. The observed name includes information about how the guard selects a name; it is not merely the event that B remains. With general release priors ${\pi }_A,{\pi }_B,{\pi }_C$, the same calculation gives ${\pi }_C/(s{\pi }_A+{\pi }_C)$, making explicit where the equal-prior convention enters.