The Three Prisoners Problem

This is an interesting topic, but the answer is not quite straightforward. The problem is stated as following:

Three prisoners. Three prisoners A, B, and C, are on death row. The governor decides to pardon one of the three and chooses at random the prisoner to pardon. He informs the warden of his choice but requests that the name be kept secret for a few days.

The next day, A tries to get the warden to tell him who had been pardoned. The warden refuses. A then asks which of B or C will be executed. The warden thinks for a while, then tells A that B is to be executed.

Warden's reasoning: Each prisoner has a 1/3 chance of being pardoned. Clearly, either B or C must be executed, so I have given A no information about whether A will be pardoned.

A's reasoning: Given that B will be executed, then either A or C will be pardoned. My chance of being pardoned has risen to 1/2.

Actually the warden's reasoning is correct. If we define A, B, C to be the events that A, B, or C will be pardoned, and W to be the event that the warden says B will die, then we can calculate P(A|W)=1/3. (The details of this calculation can be found in Statistical Inference 2ed, Section 1.3). But why is that intuitively? The warden, at first glance, seems to have given A some information about the government's decision. Where's that info gone?

To answer this question, first notice the difference between the events that the warden says B will die, and B will die. In addition to telling A that B will be executed, what the warden says in fact contains extra information: If A is to be pardoned, the warden may either tell A that B or C is going to be executed; If A is to be executed, the warden has no choice but to tell A that B is to be executed (since C is definitely pardoned in this case). So why does the warden tell A that B will die? It's more probably because that A is to be executed. Therefore, though it seems that the probability that A is pardoned seems to have risen to 1/2, the extra information, coming from the warden's choice of telling A which one of B and C will die, decreases the probability back to 1/3.

Here is a variant of the Three Prisoners Problem.

There are 3 doors, behind one of which is there a car. A contestant is asked to pick a door randomly so that he has a chance to win the car. The contestant picks door 1. Then he's shown that there's nothing behind door 3. Now should the contestant switch to door 2 to maximize the chance of winning a car? (Answer: yes)