Problem 24. The prisoner's dilemma. The release of two out of three prisoners has been announced, but their identity is kept secret. One of the prisoners considers asking a friendly guard to tell him who is the prisoner other than himself that will be released, but hesitates based on the following rationale: at the prisoner's present state of knowledge, the probability of being released is 2/3, but after he knows the answer, the probability of being released will become 1/2, since there will be two prisoners (including himself) whose fate is unknown and exactly one of the two will be released. What is wrong with this line of reasoning?
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Initially, there are three prisoners, and the probability of any individual prisoner being released is 2/3. This is because two out of the three prisoners will be released, and each prisoner has an equal chance of being one of the two. Show more…
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Three prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other two are to be freed. Prisoner $A$ asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he already knows that at least one of the two will go free. The jailer refuses to answer this question, pointing out that if $A$ knew which of his fellow prisoners were to be set free, then his own probability of being executed would rise from $\frac{1}{3}$ to $\frac{1}{2}$ because he would then be one of two prisoners. What do you think of the jailer's reasoning?
Conditional Probability And Independence
Problems
Three prisoners, A, B, and C, are held in separate cells. Two are to be executed. The warder knows specifically who is to be executed and who is to be freed, whereas the prisoners know only that two are to be executed. Prisoner A reasons as follows: my probability of being freed is clearly 1/3 until I receive further information. However, it is clear that at least one of B and C will be executed, so I will ask the warder to name one prisoner other than myself who is to be executed. Once I know which of B and C is to be executed, either I will go free or the other, unnamed prisoner will go free, with equal probability. Hence, by asking the name of another prisoner to be executed, I raise my chances of survival from 1/3 to 1/2. Investigate A's reasoning. [Hint: find the conditional probability that A is freed, given that the warder names B to be executed.]
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Please answer the attached question.
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