Concept explainers
Three Prisoners The famous “problem of three prisoners” is as follows. Three men, A, B, and C, were in jail. A knew that one of them was to be set free and the other two were to be executed. But he didn’t know who was the one to be spared. To the jailer who did know, A said, “Since two out of the three will be executed, it is certain that either B or C will be, at least. You will give me no information about my own chances if you give me the name of one man, B or C, who is going to be executed.” Accepting this argument after some thinking, the jailer said “B will be executed.” Thereupon A felt happier because now either he or C would go free, so his chance had increased from 1/3 to 1/2. Source: Cognition.
- (a) Assume that initially each of the prisoners is equally likely to be set free. Assume also that if both B and C are to be executed, the jailer is equally likely to name either B or C. Show that A is wrong, and that his probability of being freed, given that the jailer says B will he executed, is still 1/3.
- (b) Now assume that initially the probabilities of A, B, and C being freed are 1/4, 1/4, and 1/2, respectively. As in part (a), assume also that if both B and C are to be executed, the jailer is equally likely to name either B or C. Now show that A’s probability of being freed, given that the jailer says B will be executed, actually drops to 1/5. Discuss the reasonableness of this answer, and why this result might violate someone’s intuition.
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