The Quantum Monty Hall Problem is a twist on the classic and now well-known probability conundrum, where there are three doors, one of which has a car behind them. The general point is to demonstrate the counterintuitive fact that if you choose a door, and another door is revealed not to have the car behind it, you should always switch doors when the opportunity is given. The chance appears to be half and half, but really it is two-thirds and one-third.
The Quantum Monty Hall Problem goes one step farther, by applying the Heisenberg Uncertainty Principle to the game show. Instead of being told which door doesn't have a car behind it, a question is given: "Do you want to know the car's position, or how fast it's going?" Naturally, anybody on a game show would probably say, "Position!" after which they, presumably, open the door with the car behind it and get run over.