A king announces to release two out of three prisoners (A, B and C) and execute

Question

A king announces to release two out of three prisoners (A, B and C) and execute the third one, but keeps their identities secret. One of the prisoners, prisoner A, considers asking a friendly guard to tell who is the prisoner other than himself that will be released. However, he hesitates because of the following rationale: Based on his present state, his probability of being released is 2/3. But if 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 prisoner A's argument?

Explanation / Answer

We can solve this problem by taking 3 cases - each case representing one of the three prisoner who will be executed. This is done as:

In the above table the first column represents the cases - prisoner who would be executed. Now if we look at the first case where A is the one who will be executed, clearly the guard can take anyone's name - B or C to A as the person who will not be executed because in this case both B and C wont be executed. But in second case when it is B to be executed, then guard wont take B's name but will take C name with a probability of 1. Similarly we saw the third case as well where C is to be executed.

Now we will find the probability that guard will take the name of C, this is computed as:

P(C) = P( C | execute A)P(execute A) + P( C | execute B)P(execute B) + P( C | execute C)P(execute C)

Here P( C | execute A) means the probability that guard takes name of C given he knows that A would be executed. Also note that all three prisoners are intially equally likely to be executed. Therefore P(A executed ) = P(B executed ) = P(C executed ) = (1/3) Therefore now putting all the values we get:

P(C) = P( C | execute A)P(execute A) + P( C | execute B)P(execute B) + P( C | execute C)P(execute C)

P(C) = (1/3) ( 0.5 + 1 + 0) = 0.5

Similarly we will also get: P(B) = 0.5

Therefore guard is equally likely to take the name of B or C. Now we have to find the probability that given guard take one name ( either B or C ), probability that A will be executed. This is represented as:

P( A executed | C ) = P( C | execute A)P(execute A) / P(C) = 0.5*(1/3) / 0.5 = (1/3)

Therefore we see that:

P( A executed | C ) = P( A executed | B ) = 1/3

Therefore whatever guard says, the probability of A being executed will still remain (1/3) and not (1/2)

Prisoner to be Executed Prob that Guard Says B Prob that Guard Says C A 0.5 0.5 B 0 1 C 1 0

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.