The Three Prisoner’s Problem

What is the Three Prisoner’s Problem?

Three prisoners, Alex, Brian, and Cathy, 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. Alex begs the warden to let him know the identity of one of the two who are going to be executed. “If Brian is to be pardoned, give me Cathy’s name. If Cathy is to be pardoned, give me Brian’s name. And if I’m to be pardoned, secretly flip a coin to decide whether to name Brian or Cathy.”

The warden tells Alex that Brian is to be executed. Alex is pleased because he believes that his probability of surviving has gone up from 1/3 to 1/2, as it is now between him and Cathy. Alex secretly tells Cathy the news, who reasons that Alex’s chance of being pardoned is unchanged at 1/3, but she is pleased because her own chance has gone up to 2/3.

Which prisoner is correct?

Knowing that Brian is to be executed, now it’s either Cathy or Alex that will be set free. Most of us would say that Alex is correct and he’s got a 1/2 chance at freedom.

However, the correct answer is that Alex’s chances stay at 1/3 but Cathy’s changed to 2/3. With this knowledge, Cathy has doubled her chances at freedom.

How?

Though this question looks like a symmetric problem, Alex’s request to the warden made this question asymmetric. At the start, Alex has a 1/3 chance at freedom. Either way, he was going to hear Brian or Cathy’s name, he phrased his request in a way he’ll never hear his name. Thus, his chances stayed the same at 1/3. On the other hand, there was a chance that Cathy would hear her name, but she didn’t. She can reason that Alex had a 1/3 chance, but now Brian has a 0/3 chance, which means she has a 2/3 chance at freedom.

Notes

[1] The Three Prisoner’s problem is similar in structure to the more well known, Monty Hall Problem.