# Intuition and the Monty Hall problem

The Monty Hall problem goes like this: You are at a game show in front of 3 doors. There is a car behind one door and goats behind the other two doors. After you have made your choice the show master opens one of the remaining two doors, namely one with a goat. You now have the chance to change your initial choice. Do you stick to your door? Do you change? Does it make any difference at all?

The thing is, it does make a difference. If you stick, you have a probability of 1/3 to win the car, but if you change, your chances go up to 2/3. The idea is that the showmaster conveys information about where the car is by opening a wrong door. There are mathematical arguments as to why you should change. If you find that unintuitive, you are in good company. Why should you be smarter after the showmaster has opened a wrong door than before? An intuitive explanation of why the showmaster conveys information is this:

Imagine there were not 3 but 1000 doors to begin with. There is still one car but now there are 999 goats. You get to choose a single door initially. Now the showmaster does not open 1 door but 998 doors, all with goats, and you are again left with your initial choice and one more door. Do you stick to your choice now or do you change to the door the showmaster did not open? He clearly gave you a hint by not opening that particular door.