Question: Years later, there are three other prisoners, also named A, B, and C. They all know that two of them are about to be set free and that the third is to be kept in prison. Prisoner A is in his cell, pondering which one will be kept in prison. Because their crimes are different and prisoner A believes that this has been taken into consideration, he assigns different probabilities that each will be the one kept in prison. To prisoner C, an inside trader, prisoner A assigns probability 1 /2; to prisoner S, a check forger, he assigns probability 3/8; and to himself probability 1/8.
Being curious, prisoner A asks the warden to tell him which of the other two prisoners is to be set free. The warden agrees to tell the truth.
a. Let p be prisoner A's probability that the warden will say "B goes free," given that both B and C are to be released. For what value of p will the warden's response be irrelevant to A's beliefs about whether he himself will stay in prison?
b. Now suppose that, given both B and C are to be set free, prisoner A believes there is a 1/3 chance the warden will say "B goes free;" otherwise, he'll say "C goes free."
1. According to A, what is the probability that the warden will say that B goes free? That C goes free?
2. Suppose the warden tells prisoner A that B will go free. Now what should prisoner A assign as the probability that he'll stay in jail, given the warden's statement?
3. Suppose instead that the warden says that C will go free. What should prisoner A assign as the probability that he'll stay in jail?