Monty Hall problem facts for kids

The Monty Hall problem is a famous puzzle about probability, which is the study of chance. This problem comes from an old American TV game show called Let's Make a Deal. It's named after the show's host, Monty Hall.

Imagine you are on this game show. There are three closed doors. Behind one door is a cool car, which is the big prize! Behind the other two doors are goats, which are not as exciting.

First, you pick one door, but you don't open it yet. The host, Monty Hall, knows exactly what's behind every door. He then opens one of the other two doors, and he always picks a door with a goat behind it. He will never open the door you chose, and he will never open the door with the car.

After Monty opens a goat door, he gives you a choice: Do you want to stick with your first choice, or do you want to switch to the other closed door? The big question is: Does switching doors give you a better chance to win the car?

What is the Monty Hall Problem?

The Monty Hall problem is a brain teaser that shows how our minds can sometimes trick us about chance. Most people think that after one goat door is opened, the remaining two closed doors each have a 50/50 chance of hiding the car. But that's not quite right!

The Game Rules

Here are the simple rules of the game:

• There are three doors.
• One door has a car, and two doors have goats.
• You pick one door.
• The host, who knows where the car is, opens one of the other doors that he knows has a goat.
• The host then asks if you want to switch your choice to the remaining closed door.
• The host must open a goat door.
• The host must offer you the chance to switch.

Why Switching is Smart

It might feel like the chances are 50/50 after one door is opened, but they are not. The amazing truth is that if you switch doors, your chances of winning the car go up! They go from 1 out of 3 (1/3) to 2 out of 3 (2/3).

This happens because of how the game starts. When you first pick a door, you have a 1/3 chance of picking the car. This means there's a 2/3 chance that the car is behind one of the other two doors.

When the host opens a goat door, he's actually giving you a big clue. He's concentrating that 2/3 chance onto the one remaining closed door. So, by switching, you're essentially choosing the group of two doors that had a 2/3 chance of having the car, instead of sticking with your original single door that only had a 1/3 chance.

How Switching Helps You Win

Let's look at the three possible ways the game can play out. Remember, you always pick a door first, and the car can be behind any of the three doors.

1. Host reveals Goat A or Host reveals Goat B Player picks car Changing loses. 2. Host must reveal Goat B Player picks Goat A Changing wins. 3. Host must reveal Goat A Player picks Goat B Changing wins.

The player has an equal chance of initially selecting the car, Goat A, or Goat B. Switching results in a win 2/3 of the time.

Here are the three possibilities:

• Scenario 1 (You Lose if you Switch):

* You pick the door with the car (1/3 chance). * The host opens one of the other two doors, which must have a goat. * If you then switch, you will get the other goat. So, you lose.

• Scenario 2 (You Win if you Switch):

* You pick a door with a goat (1/3 chance). * The host must open the other goat door. * If you then switch, you will get the car! So, you win.

• Scenario 3 (You Win if you Switch):

* You pick the other door with a goat (1/3 chance). * The host must open the first goat door. * If you then switch, you will get the car! So, you win.

As you can see, in two out of three possible situations, switching your choice leads to winning the car. This means if you always switch, you will win the car two times out of three!

See also

• In Spanish: Problema de Monty Hall para niños