The Monty Hall Problem

The Monty Hall Problem 🎮

You’re on a game show, and there are three doors in front of you:

Behind one door is a car.
Behind the other two doors are goats.

You pick one door, and then the host (who knows what’s behind each door) opens another door, revealing a goat. He then gives you the opportunity to switch your choice to the remaining unopened door.

Should you stick with your original choice or switch? Does it matter?

Hint:

When the host opens a door and reveals a goat, that actually provides you with additional information about the game. Initially, each door has a 1 in 3 chance of being the car, but after the host reveals a goat, the probability shifts.

Think about this:

When you initially pick a door, there’s a 1/3 chance it’s the car and a 2/3 chance the car is behind one of the other two doors.
After the host reveals a goat behind one of the other doors, the 2/3 chance that the car was behind the other two doors is now concentrated on the remaining unopened door.

Answer:

The Key Insight:

When you first pick a door, you have a 1/3 chance of picking the car and a 2/3 chance of picking a goat. The host, knowing where the car is, always opens a door that has a goat behind it. This changes the probabilities once the host reveals a goat behind one of the other doors.

The Strategy:

If you stick with your original choice, your chance of winning the car remains 1/3 because that was the initial probability of your first choice being the car.
If you switch to the remaining door after the host reveals a goat, you have a 2/3 chance of winning the car. This is because, when you made your initial choice, there was a 2/3 chance the car was behind one of the other doors, and by switching, you’re effectively betting that your first pick was wrong.

Conclusion:

You should always switch. The probability of winning by switching is 2/3, while the probability of winning by sticking with your initial choice is only 1/3.

Why Does This Work?

Initial choice: There’s a 1/3 chance the car is behind the door you initially pick.
Switching: There’s a 2/3 chance the car is behind one of the other two doors, and when the host reveals a goat behind one of them, you’re left with the remaining door, which has a 2/3 chance of being the car.