The Monty Hall Problem

How you play this game determines your chances of winning, but you may be surprised to learn what the best strategy is.

This math puzzle is named after Monty Hall, the host of an American television game show called Let’s Make a Deal, which was on during the 1960s and 1970s. In one of the games, Monty presents contestants with three doors.

Behind one of the doors is a car. Behind each of the other two doors is an empty room. Monty knows what’s behind each door, but you don’t.

The game is played in three steps:

You get to pick a door.
Monty opens one of the two doors you did not pick to reveal an empty room. (He never opens the door with the car behind it.)
You now have a choice to either remain with the door you picked in step 1 or to switch to the other door that is still shut.

Let’s say you pick door A. Monty opens one of the other two doors: let’s say B.
Now you have a choice to switch to C or stay with your original choice, A. If you stay, you may be lucky . . .
or not.

On the other hand, if you switch to C, you may be lucky . . . or not.

What do you do? Do you stay with your original choice or switch after Monty opens a door. Why?

Course:

Math [1]
Probability [2]

Result/Solution(s)

Solution: The Monty Hall Problem Math Puzzle

It pays to switch. Your chances of winning are twice as good as when you stay with your original choice. This surprises many people. You’ve got a 1/3 chance of picking the door with the car behind it. How can Monty’s opening another door make any difference? The car hasn’t moved.

Here’s why it’s better to switch:
If you pick door A, you have a 1/3 chance of winning, since the probability of the car being behind door A is 1/3. The probability of the car being behind door B is 1/3, and the probability of it being behind door C is also 1/3. (The probabilities have to add up to 1, since the car is certain to be somewhere.) The probability of the car being behind door B or door C is 2/3.

Now let’s say that Monty opens door B to reveal that it is empty. The probability of the car being behind door B or door C is still 2/3, but we now know that the probability of it being behind door B is 0, since it’s certainly not there. So the probability of it being behind door C is now 2/3. The probabilities still add up to 1: 1/3 for A, 0 for B, 2/3 for C.

Still not convinced? Try this thought experiment: There are 1,000,000 doors. You pick one of them hoping for the car. You’ve got a million in 1 chance of being right. The chance that the car is behind one of the other doors is 999,999 out of a million. Monty opens 999,998 doors to show that they are empty. Your original guess had a 1 in a million chance of being right. If you were wrong, switching gets you the car for sure. Do you switch?

Still not convinced? Try the game with a friend. Use three paper cups and a little toy car, or some other object. One of you should pretend to be Monty hiding the car and lifting the empty cup after the other player makes a choice. Play about 100 times and see what you come up with.

Still not convinced? Go to the Web site Monte Hall Paradox [3] and try it for yourself. Be sure and select “host knows”; that is, Monty knows what’s behind the doors, which is the way our puzzle is presented.

Probability puzzle [4]
Math Puzzle [5]