The Monty Hall puzzle is a probability problem developed after the game show „Let’s make a deal“, and named after its host, Monty Hall.
The puzzle goes like this:
There are three closed doors in front of you. Behind two doors there’s a hidden goat, and behind the third one there is a valuable prize (a car, a suitcase filled with money, whatever you please, but we’ll stick to the car in this article). The game show host tells you to pick one door. Remember that the host knows which door hides the valuable prize.
After you’ve made your choice, the host will open one of the two remaining doors, and behind it there will be a goat. That means that behind the two closed doors there must be a car. Then the host asks you if you want to swap your chosen door or stick to your initial pick.
The question is: will there be a difference if you change your choice?
Most people will say there will be none, because the chances of winning a car will be at 50%. However, that is not correct.
Let’s analyze why the common assumption that there will be no difference is incorrect:
- At the beginning, before the host opened one of the doors, you had a roughly 33% chance of choosing the door with the car, and 66% of winning a goat.
- When the game show host opened one door, the chances remained the same. Why? Because at the beginning you had 66% chance of choosing a door with a goat. It is more likely that the door you had chosen was hiding a goat than a car.
- If you stuck to the inital choice, you would have 66% chance of winning the goat. If you were to swap the chosen door, the chances would change and you would then have a 66% chance of winning the car!
Below is a table that gives you all the possible outcomes.
Napisala: Mateja Napravnik