The Monty Hall problem is famously unintuitive. Recently, people have tried to give equivalent problems where the solution seems intuitive.

I have my own way of explaining this. I’ll start with a different game where the answer is “obvious”. Then, I’ll make tiny changes that don’t really matter.

GAME 1

Here’s the game:

There are 100 doors. A car is randomly placed behind one, and goats behind the other 99.

You pick one door.

Then, you get two options: Option A: You get whatever is behind the door you picked. Option B: You get whatever is behind all of the other 99 doors.



If you (like me) love goats more than cars, put that aside. Goats have zero value and cars high value.

For this game, you should obviously choose option B. There’s only a 1% chance you picked the right door, so there’s a 99% chance that the car is behind one of the others.

GAME 2

Here, the game is slightly updated: (new part in bold)

There are 100 doors. A car is randomly placed behind one, and goats behind the other 99.

You pick one door.

Monty says “Hey! I promise you that there is a goat behind at least 98 of the other 99 doors!”

Then, you get two options: Option A: You get whatever is behind the door you picked. Option B: You get whatever is behind all of the other 99 doors.



What should you do? Monty’s statement doesn’t give you any new information. You don’t need to rely on his trustworthy looks— you already knew there were at least 98 goats. So, this is no different from GAME 1. The correct decision is still option B, which gets you the car 99% of the time.

GAME 3

Now, let’s slightly update the game again: (new part in bold)

There are 100 doors. A car is randomly placed behind one, and goats behind the other 99.

You pick one door.

Monty looks behind the other 99 doors. He chooses all but one of them to open, revealing 98 goats.

Then, you get two options: Option A: You get whatever is behind the door you picked. Option B: You get whatever is behind all of the other 99 doors.



The key insight is that Monty showing you that 98 of the other 99 doors contain goats has not given you any information. You already knew there were 98 goats! So this is just like the previous games. The correct decision is still to switch, which still gets you the car 99% of the time.

This is the critical insight: Imagine Monty walking down the doors, opening them one by one, but skipping a single door. Doesn’t that door seem special to you?

GAME 4

Finally, we arrive at a game very much like Monte Hall.

There are 100 doors. A car is randomly placed behind one, and goats behind the other 99.

You pick one door.

Monty looks behind the other 99 doors. He chooses all but one of them to open, revealing 98 goats.

Then, you get two options: Option A: You get whatever is behind the door you picked. Option B: You get whatever is behind the other closed door.



The only difference with GAME 3 is that option B doesn’t get you the 98 visible goats. Since you don’t care about goats, this makes no difference. This is still exactly like the other games. You get the car 99% of the time by switching.

GAME 5:

Here is the last game. We now have 3 doors instead of 100.

There are 3 doors. A car is randomly placed behind one, and goats behind the other 2 .

doors. A car is randomly placed behind one, and goats behind the other . You pick one door.

Monty looks behind the other 2 doors. He chooses one one of them to open, revealing 1 goat.

doors. He chooses one one of them to open, revealing goat. Then, you get two options: Option A: You get whatever is behind the door you picked. Option B: You get whatever is behind the other closed door.



Of course, you still want to choose option B. The probability of success is now 2/3 instead of 99/100.

Side note: It’s important that Monty looked behind the doors before choosing which open. This is where people’s intuition usually fails them. If he had chosen a door at random — in a way that he risked possibly exposing a car, then that would give you some information. But he doesn’t choose the door at random. He deliberately chooses to show you goats. Since this is always possible, it tells you nothing. I think this is the crux of what makes this problem unintuitive – people are intuitively reasoning as if the door was opened at random.

Extra side note: Did you know Monty Hall was actually named “Monte” at birth?