I am given an exercise, and I can't quite figure it out.
The Prisoner ParadoxThree prisoners in solitary confinement, A, B and C, have been sentenced to death on the same day but, because there is a national holiday, the governor decides that one will be granted a pardon. The prisoners are informed of this but told that they will not know which one of them is to be spared until the day scheduled for the executions.
Prisoner A says to the jailer “I already know that at least one the other two prisoners will be executed, so if you tell me the name of one who will be executed, you won’t have given me any information about my own execution”.
The jailer accepts this and tells him that C will definitely die.
A then reasons “Before I knew C was to be executed I had a 1 in 3 chance of receiving a pardon. Now I know that either B or myself will be pardoned the odds have improved to 1 in 2.”.
But the jailer points out “You could have reached a similar conclusion if I had said B will die, and I was bound to answer either B or C, so why did you need to ask?”.
What are A’s chances of receiving a pardon and why? Construct an explanation that would convince others that you are right.
You could tackle this by Bayes theorem, by drawing a belief network, or by common sense. Whichever approach you choose should deepen your understanding of the deceptively simple concept of conditional probability.
Here's my analysis:
This looks like the the Monty Hall problem, but not quite. If A says
I change my place with B after he is told C will die, he has 2/3 chances to be saved. If he doesn't, then I would say his chances are 1/3 to live, like when you don't change your choice in the Monty Hall problem. But at the same time, he is in a group of 2 guys, and one should die, so it is tempting to say that his chances are 1/2.
So the paradox is still here, how would you approach this. Also, I have no idea how i could make a belief network about this, so i'm interested to see that.