$3\mathrm{.}1$ Three prisoners, A$,$ B$,$ and C$,$ are held in separate cells. Two are to be executed. The warder knows specifically who is to be executed, and who is to be freed, whereas the prisoners know only that two are to be executed. Prisoner A reasons as follows: my probability of being freed is clearly $13$ until I receive further information. However, it is clear that at least one of B and C will be executed, so I will ask the warder to name one prisoner other than myself who is to be executed. Once I know which of B and C is to be executed, either I will go free or the other, unnamed, prisoner will go free, with equal probability. Hence, by asking the name of another prisoner to be executed, I raise my chances of survival from $13$ to $12.$ Investigate A$$s reasoning. $[$Hint: find the conditional probability that A is freed, given that the warder names B to be executed.