Suppose you're at a game show, knowing the winning strategy for the classic Monty Hall problem. Your turn comes up and you take the stage. You confidently choose door number (?).
Just as you are getting ready to tell the host you want to switch doors, the host announces that the rules have been changed for today's show. You groan in disappointment.
The host explains that no doors are going to be revealed or opened prematurely, unlike the classic Monty Hall problem. Instead, you are given the choice of keeping the door you picked, or switching to the other two doors. If you stay, and the car is not behind your door, you lose. If it is, you win. If you switch, and the car is behind one of the two other doors, you win. It can be behind either one. If it was behind the first door you picked, you lose.
Suddenly you feel confident again. You realize that just like in the classic Monty Hall, your chances of winning the car if you switch are 2/3: you now get two doors instead of one. You tell the host you would like to switch to the two doors.
The host then asks you to reconsider, explaining that your chances aren't quite as good as you thought. The odds of a single door hiding a goat are 2/3, since two of the three doors hide a goat. Then, multiplying these together (for two doors) yields a probability of 4/9 that both of your two doors contain goats, yielding you a 5/9 chance of winning the car. The host grins as you realize your anticipated 2/3 odds have shrunk slightly, but you still decide to switch to the two doors. After all, 5/9 is still better than 1/3.
As you leave, goat leashes in hand, it still bothers you that your initial intuitive guess seemed to have been proven wrong by the host. You console yourself with the fact that you won both goats. Who ever needed a lawn mower, anyhow?
What is your true probability of winning if you switch to the two doors? Who was right, and why?
Bonus point challenge: There is yet another way to mathematically prove which is the correct answer, found by dark wanderer (using the sum of all probabilities rule).