Monty hall is a notoriously difficult problem to understand. When it was first published, thousands of people called the newspaper to tell them that the assertions is wrong including multiple PhDs.
If you already know the problem, please skip to the solution in the second section.
What is the problem?
In the monty hall game, a host shows three doors to the player. Behind one of the doors is a car (win) and other two hide a goat (loss). The host knows what’s behind each door.
The host asks the player to pick one of the doors. When the player picks one of the doors, the host doesn’t reveal the chosen door instantly. Instead they open one of the doors which the player didn’t choose, reveal a goat and ask the player if they want to switch their choice.
Now the monty halls problem says that switching is always better.
The common intuition is that when the player picks the door the first time, they had 1/3 (33%) chance of being right. When the host eliminated one of the door, then the player had 1/2 chance of being right and it doesn’t matter if they switch or not.
I’ve seen multiple explanations from different people but I wasn’t able to grasp the reason despite mathematical explanations.
Intuitive way of thinking about the problem
Its possible to understand the monty hall problem without any maths or probability and I think its the most intuitive way.
Stop thinking about it from the player’s perspective and imagine yourself as the host.
The player has just picked one out of three doors.
Now the decision is yours and you have two doors.
As a host, you cannot pick the door with a car.
Since you have two doors to choose from, the chance that you will get a door with a car is greater.
Now because you can only to open the door with goat, you will always avoid the car.
That’s the key to this problem!
As soon as you open the door with a goat, you non-verbally tell the player:
Hey I know where the car is.
I have greater chances of getting a car than you.
And I am avoiding this door out of the two I need to pick from.
That information changes the game for the player. They should pick the door you avoided every time because you have more chances of getting the car than him and you cannot open the car one.