Suppose you’re on a game show and there are three doors, behind two of which there is a goat, and the remaining one a car. Your objective is to guess which door has the car. You’re allowed two guesses except after the first guess the host of the game show (who knows exactly which door has a goat and which has a car) will open one of the other two doors, always revealing a goat. You’re now given the opportunity to switch doors. The question is what should you do? Switch, or stick with the same door as your first choice (may be because you think the host is trying to play reverse psychology on you). If you’re a probability enthusiast then you’ll know that this is the well-known Monty-Hall problem, and if you’re a not, then (I hope) that you think the answer is very intuitive – namely that it doesn’t make any difference whether you switch doors or not, the probability of choosing the door with the car is 50%. Well, when I first met this problem some years ago I learnt that this intuition is wrong. It completely shattered any confidence I had in probabilities. As a graduate student I learnt what the answer* is and understood the reasoning behind it but the Monty-Hall problem still bothers me somewhat even now. It still bothers me not because I have trouble recalling the solution, but because I found it difficult to resist the temptation to follow the 50/50 intuition on a recent revisit of the problem. It seems that most of us have trouble resisting this temptation (which is why the Monty-Hall problem became famous). To me this means that we just don’t have a robust intuition about probabilities. As the famous polymath John von Neumann said “in mathematics you don’t understand things. You just get used to them.”
*I’m not going to tell you the reason why this is wrong (because it’s readily available on the internet and also because I don’t want to spoil the fun of solving the problem for those who like to challenge).