I had this problem beaten to death in my probability and statistics courses in college. To make the problem much more obviously intuitive, simply imagine a tho

I had this problem beaten to death in my probability and statistics courses in college. To make the problem much more obviously intuitive, simply imagine a thousand or a million doors instead of 3.. and after you make your pick - the host reveals all doors except one to be "failures". Your choice was very obviously 1 in a million, and because the reveal procedure that followed is conditional - if the "success" was behind ANY of the doors you did not pick then it WILL be the only door left in front of you. It is only a 50-50 IF the revealed doors were opened randomly. In this case, sometimes the "success" would be revealed by accident - and so IF you do happen to find yourself in a position of only having 2 doors left THEN it was more likely to happen if you had actually chosen the "success". So arriving at that situation at all is the conditional information that alters the probability of your original choice - but in this case from 1/3 to 1/2 instead of 2/3. (this is how the probabilities in "deal or no deal" work btw) Even knowing the solution, it is a good reminder about the power of information derived from systemic processes. the information leveraged in the processes that lead to circumstances is just as important as the circumstances themselves. The key general point that bypasses most peoples intuition the first time they hear the problem is that the revealed box (or door, etc) is not chosen at random but by a conditional process. The condition is that the host reveals a "failure". In order to pick a failure, he must leverage knowledge about where the "success" is, else he risks revealing it by mistake. Because knowledge of the "success" is embedded into the decision process, more information about the "success" is revealed to the observer which allows them to improve their position.