The Monty Hall problem explanation

Have a go at the problem. Does it matter whether you change your mind or not?

To repeat the problem: There are three doors, two with goats behind and one with a car. You hope to choose the door hiding the car. When you have chosen, the door isn't opened, but the host opens one of the other doors, showing a goat. (It is essential for the problem that the host knows which door hides the car, that he doesn't reveal the car, and that he doesn't open the door that you've chosen.) Now you have the chance to change your mind. There are two doors left. Should you stay with your original choice, or should you chose the the last door instead?

If you have tried the problem, you should find that if you stick with your first choice, you seem to get more goats than cars. But if you change your mind, you start 'getting lucky' and get more cars than goats. (If this hasn't happened, go back and play it several more times. Make sure that you do about 20 goes each time.)

This is a probability problem and these can be confusing. The first thing to remember is that probability never tells you what is going to happen. It merely gives you odds for something happening or not. You feel intuitively that it shouldn't matter whether you change your mind or not. After all, the car and goats are going to stay where they are! But this is looking at it the wrong way. If you don't change your mind, you are only using the information that you had to start with. There are three doors, hiding two goats and one car. The chance of getting the right door is one in three, or a third. You ignore the games host, who is in fact giving you some more information. So the odds stay at a third. That means that you will tend to get more goats than cars, and if you play it enough, for long enough, you will see that this usually happens.

So what happens if you change your mind? Well, I mention in the description that the host knows where the car is, and he is careful to open the door with a goat, not the car door. This means that he has given you some useful information. How can you use it? It's best for simple probability to list all possibilities. Let's call the doors 1, 2 and 3, and assume that you have chosen door 1. What are the different outcomes?

• The car was behind door 1 (so you were right to start with). The host opens door 2 or door 3 (it doesn't matter which). You change your mind, and that means you chose another door than 1, which will have a goat, so you lose.
• Door 1 has a goat. Door 2 has the car. That means that the host has to open door 3, with the other goat. You change your mind, and chose door 2. This has the car, so you win.
• Door 1 has a goat. Door 3 has the car. That means that the host has to open door 2, with the other goat. You change your mind, and chose door 3. This has the car, so you win.

So in the three cases, if you change you mind, you lose once, and win twice. That means you get more cars than goats. So it pays to change your mind!