The Monty Hall Problem is a fantastic probability brain teaser based on the American television game show Let's Make a Deal—and this video is the best explanation of it you're likely to find.
The problem is simple. You're on a game show, and you're given the choice of three doors: behind one door is a car; behind the others, goats. You choose one door—say, number 1—and the host (who knows what's behind the doors) opens another door—say number 3—which has a goat behind it. You're then allowed to stick with the door you picked, or choose the other one—number 2 in the example.
The question is: should you switch your choice? The answer is simultaneously blindingly simple and fiendishly counterintuitive. But the above video might be your best chance yet at wrapping your head around it. [YouTube]