Question: Three prisoners-A, B, and C-are in a prison camp. They all know that one of them is to be executed and the other two are to be set free. However, the warden has not announced who will be executed. Being curious, prisoner A asks the warden, who is an honest man, to tell her which of the other two prisoners is to be set free; she already knows that at least one of the other two will be set free.
a. Suppose that prisoner A believes that it is equally likely that any one of the three prisoners will be executed. She also thinks that if both prisoners B and C are to be set free, the warden is equally likely to name either one. The warden quietly tells her that prisoner B is to be set free and she agrees not to tell this to the other two prisoners. In light of the warden's remarks, what probability should prisoner A assign to the event of being executed herself? Does this probability make sense to you? (Adapted from Frederick Mosteller, Fifty Challenging Problems in Probability, Dover, New York, 1965.)
b. Prisoner C believes that he is twice as likely as B to be the one executed, and B and A are equally likely to be the unfortunate one. He asks the warden the same question that A asked. He believes that if both prisoners B and A are to be set free, the warden is twice as likely to name A as B. If the warden tells him B is to be set free, what probability should prisoner C assign to the event of A being executed?
c. A new prisoner, named D, enters the camp and prisoner A tells him that either prisoner A (herself) or prisoner C will be executed. Prisoner D believes that the warden favors neither A nor C. What probability should prisoner D assign to the event of prisoner A being executed? Compare this to your answer for part (a) and explain.