Conditional Probability Andy, Bob and Charley have all been serving time for grand theft auto.  The warden plans to release two of them next week at random.  However Andy is friends with one of the guards who offers to tell Andy the name of one of the prisoners being released other than himself at random.  Andy declines as he believes that if he had that information then his chances would decrease to $\frac{1}{2}$ instead of $\frac{2}{3}$.  Is Andy's concern justified? 
This Problem seems obvious that his concern isn't justified, i'm just having trouble proving that's the case.  What I thought is that I would try and show that $$P(A \mid B \cup C)=\frac{2}{3}$$
Where $A$ is the event that Andy gets released $B$ is the event that Bob gets released, and $C$ is the event that Charley gets released.  
 A: This is a variation on the standard Monty Hall problem in that the jailer has full knowledge and chooses accordingly. We'll add the assumption that in the event $B,C$ are both being released the jailer chooses which to name uniformly at random.
At the start the state space has three elements $AB,AC,BC$ each with probability $\frac 13$.  Let $X$ be the event "the jailer says $B$".  If the jailer says $B$ then Andy knows that $AC$ is impossible.  But the probability that the true state is $AB$ must be calculated via Bayes Theorem.  We see that $$P(AB\,|\,X)=\frac {P(X\,|\,AB)\times P(AB)}{P(X\,|\,AB)\times P(AB)+P(X\,|\,BC)\times P(BC)}=\frac {1}{1+\frac 12}=\frac 23$$
Thus Andy learns nothing about himself from whatever the jailer says.
Intuitively:  This is not surprising.  In all states of the world the jailer will be able to name someone so the fact that he can name someone conveys no information.
Note:  to make the problem more like the Monty Hall problem, suppose, after saying $B$ to Andy the jailer offers Andy the chance to swap cells with $C$.  The above shows that Andy should decline!  Poor $C$ is down to a $\frac 13$ chance.  Indeed, Andy did get information regarding $C$.  The fact that the jailer named $B$ and not $C$ is evidence for the statement that $C$ is not being released. Thus Andy should stay in his own cell.
