# The Three Prisoners Problem **See Also:** [[Probability]], [[Critical Thinking]]. **Source:** [Three prisoners problem](https://en.wikipedia.org/wiki/Three_prisoners_problem). >[!quote] >Three prisoners, A, B, and C, are in separate cells and sentenced to death. The governor has selected one of them at random to be pardoned. The warden knows which one is pardoned, but is not allowed to tell. Prisoner A begs the warden to let him know the identity of one of the two who are going to be executed. The warden tells A that B is condemned, and A believes their odds of survival have increased from 1/3 to 1/2. Is A correct? This problem is interesting as Brain Candy. My comment about solving it, which doesn't seem to appear in the source, is: It's not a question of calculating the odds of being pardoned, it's a question of calculating what the guard's responses can be. These are 2 different situations; in the former, no extra decisions are made, while in the latter, the guard makes choices. Alternatively, the question is not whether any new information is received by A, but whether the guard can deliver any new information at all. There are two key facts. - Prisoner A asks to know which *other* prisoner is condemned. - By the Warden's order the guard cannot tell A about A's own disposition. - This rule is often ignored by naive reasoners. Since two of the three prisoners are condemned, A *already knows* that either B or C are condemned. There is no need to ask the guard because the guard's response adds no new information. Without new information, the probabilities *cannot* change. The second fact is analogous (but leads to a different deduction) to the rule in the Monty Hall Problem that Monty *must* open a door with a goat. If A is pardoned, then G can say B *or* C. If B is pardoned, then G must say C, or A will know they are condemned. If C is pardoned, then G must say B, or A will know they are condemned. The point is that no matter what the guard says (B or C), it can cover any of the three cases. Again, no new information is provided to A, regardless of what the guard says, so the probabilities cannot change. This also emphasizes the importance of having deep [[language]] skills, because it is by carefully parsing the rules that one can deduce the solution.