[bogosmith]

The Monty hall problem is a very nice example of why reasoning with conditional probabilities is hard. If you have studied conditional probabilities and encounter the Monty Hall problem it is easy to think you are in the situation to apply your knowledge and reason like that:

I know that the prize is not behind door A (because the host showed me the goat there). Given that the prize is not behind door A calculate the probability that it is behind door B. Well that's obviously 1/2. Why? Just use conditional probabilities that's what they are for. (The calculation is trivial: Event X="prize behind B", event Y="prize not behind A", P(X)=1/3, P(Y)=2/3. P(X and Y)=1/3. P(X|Y)=P(X and Y)/P(Y) = 1/2.)

Exactly why this argument is wrong is the subtle part. Of course the argument doesn't put to use part of the information given - which door I chose originally and what contract I had with the host, so this is normally enough to make one uncomfortable, but still we know that, e.g.

"In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (...) has already occurred." (source: wikipedia)

Having thought about it I find that adding some more precision to the intuitive motivation of conditional probabilities might be helpful. I certainly think twice now before using conditional probabilities to model the real world, but realizing that I should has cost me quite a bit of head scratching.