> It’s important that Monty looked behind the doors before choosing which to open. This is where people’s intuition usually fails. If he had chosen a door at ra

> It’s important that Monty looked behind the doors before choosing which to open. This is where people’s intuition usually fails. If he had chosen a door at random — in a way that he risked possibly exposing a car, then the situation would be different. (In that case, there’s no advantage or harm in switching.) This one took a second to rationalize. The reason it works is you have the same chance of winning overall (assuming you can safely choose the car if he opens the car by chance), it's just that the value of winning from switching vs. staying has been shifted into the probability of winning by default. The usual mental trick is to extend to 1,000,000 doors. If you pick one door, then are told that all of the alternatives except one are the correct answer, you should obviously reason that the door you didn't choose is the correct answer, unless you got the 1/1,000,000 guess. Odds of winning if you switch are 999,999 / 1,000,000 If the host instead opens 999,998 doors randomly, you have a 999,998 / 1,000,000 chance of winning by default. The remaining two doors have equal chance of winning, giving you the same total odds of 999,999 / 1,000,000 no matter which you choose. Which makes sense, because both situations are, more or less, being given 999,999 chances to guess the lucky door.