# Monty Hall problem.

Suppose you're on a game show, and you're given the choice of three doors.

Behind one door is a car, behind the others, goats.

You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat.

He says to you, "Do you want to pick door #2?"

Is it to your advantage to switch your choice of doors?

The answer is "yes". The chances of winning are doubled by switching.

The odds go up from the original 1/3 to 2/3 if you switch.

This may be the simplest way to explain what is going on here.

You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat.

The probability that the car is behind door #1 is 1/3.

That means the probability that it is behind other two doors (#2 and #3) is 2/3.

The host now opens one of the doors, #2 or #3, which has a goat. Let's suppose he opens door #3.

Notice that the host knows where the car is located. (This is the crucial piece of information and is the key to solve the problem.)

You now have 3 relevant pieces of information:

1. The probability that the car is behind door #1 is 1/3.

2. The probability that the car is behind door #2 or #3 (i.e., not behind door #1) is 2/3.

3. The car is not behind door #3.

With these 3 pieces of information you can conclude that the probability that the car is behind door #2 is 2/3.

Therefore it is to your advantage to switch from the original choice of door #1 to door #2.

The reason is simple. your original guess is right with probability 1/3, whereas one of the other two choices is correct with probability 2/3.