You are on the game show Let's Make a Deal. There are three brightly colored doors labeled 1,2 and 3. Behind one of them is a brand new car, and you will win the car if you pick that door.You are standing next to the host, Monty Hall. He asks you to pick a door, and tells you that after you pick your door, he will open one of the other doors to show that it is empty, and give you the opportunity to switch your choice to the remaining closed door, or stik with your original choice. You pick door number 2, Monty opens up door number 3.
Is it better to switch to door number 1, or stick with number 2?
The result is counterintuitive, so most people answer this question incorrectly.
It is always better to switch. The most common initial response is that there is no benefit either way. The assumption is that since there are two doors, and the car could be behind either, the odds are even.
This is a mistake because it ignores the extra information gained from the opening of the empty door. In most cases (2/3 of the time) your initial choice will be an empty door, that means that the door Monty opens is the other empty door, and the door you would switch to will have the car behind it. Only 1/3 of the time will you lose by switching to the other door, if you happen to have picked the car in the first place.
The odds would be fifty-fifty, except that by opening one of the two unchosen doors, the show gives you a little extra information about the unchosen doors. Before the game starts, you know each door has a 1/3 probability of having a car. When you learn one of two unchosen doors does not contain the car, its share of probability is transferred to the remaining unchosen door. The probability of your initial choice remains unchanged at 1/3. Because it was not subject to being revealed, no information could be gained about it.