Example 1: The Curious Prisoner

There are 3 prisoners -- A, B and C, of which 2 are to be executed. Only the jailer knows which two. A, the curious, asks the jailer: ``At least one of the other two are to be executed and we both know it. So you will be giving me no useful information if you tell me who it is.'' The jailer thinks about this for a while. He then says: ``B will be executed.'' Is the posterior probability of A's fate any different?

Let X represent the event that X will escape

JX represent the event that the jailer names X as the person to be executed.

We are seeking:

P(A|JB), i.e. the probability that A will escape given that the jailer names B.

By Bayes rule:

$$\begin{eqnarray*}P(A\vert JB) &=& \frac{P(A) \cdot P(JB\vert A))}{\sum\limits_{X... ...c{\frac{1}{2}}{\frac{1}{2} + 1}\\ &=& \frac{1}{3} \quad still! \end{eqnarray*}$$