One of the doors has a sports car behind it. The other two have goats. Can you pick the door with the sports car?
When you are asked to switch, you are probably thinking, "There are two doors left, both have a 50% chance of having the goat behind them." But this is not so! ::Explanation 1:: When the game begins, there are three doors, so each door has a 1/3 chance of having the sports car. Once you have selected a door, a different door is revealed to have a goat. You are then asked if you would like to switch doors. The door which you would switch to actually has a 2/3 chance of having the car, not a 1/2. To understand why this is, imagine 100 doors. The situation is the same: one of the doors has a sports car, the other 99 have goats. You pick your door, it has a 1/100 chance of having the sports car. So it is very, very likely one of the other 99 doors has the car. 98 of the 100 doors are opened and revealed to have goats in them. Now there are two doors left, your door and the door which was not revealed. Remember that there was a 99% probability that your chosen door does not have the car and it is behind one of the other doors. Now that 98 of the "other" doors are out of the equation, there is a 99% chance that the car is behind the non-revealed door. Would you switch? Of course you would! Basically, every door that is revealed to not have the car splits its probability among the remaining doors. ::Explanation 2:: Imagine 100 doors. Same situation, one of them has the car, the other 99 have goats. Lets say you pick door #1. There is a 1% chance your door has the car and a 99% chance one of the other doors has the car. Now the other 99 doors are replaced with 1 huge door that has 99 smaller doors on it, like compartments. There is still a 99% chance of the car being in the big door. Now 98 compartments on the big door are opened and revealed to have goats. You are then asked if you would like to switch to the big door. Of course you switch because there was, and still is, a 99% chance the car is in the big door as opposed to the 1% of your original door. Hopefully you understand now why you should always switch doors.
