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 others who are going to be executed. “If B is to be pardoned, give me C’s name. If C is to be pardoned, give me B’s name. And if I’m to be pardoned, flip a coin to decide whether to name B or C.”
The warden tells A that B is to be executed. Prisoner A 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 C. Prisoner A secretly tells C the news, who is also pleased, because he reasons that A still has a chance of 1/3 to be the pardoned one, but his chance has gone up to 2/3.
What is the correct answer?
A pardoned; (B executed); C executed; possibility of this combination = 1/3 * 1/2 = 1/6
A pardoned; B executed; (C executed); possibility of this combination = 1/3 * 1/2 = 1/6
A executed; B pardoned: C executed; possibility of this combination = 1/3 * 1 = 1/3
A executed; B executed: C pardoned; possibility of this combination = 1/3 * 1 = 1/3
Total possbility = 1/6 + 1/6 + 1/3 + 1/3 = 1 but two of the combinations were impossible.
Therefore, the A to B’s ratio is (1/6) : (1/3) = 1:2, so C’s chance is actually 2/3.
Benc Tan 