An Intuitive Explanation of the Monty Hall Problem

The Monty Hall problem goes like this:

Imagine you’re on a TV game show, and you’re given the choice of three doors: Behind one door is a car; behind the other two doors are goats. You don’t know which is which. Suppose you pick Door Number 1. Now the host, Monty, who knows what’s behind the doors, always opens another door that has a goat, say Door Number 2. He then says, “Before opening door Number 1, I’m offering you the chance to switch to Door Number 3?” Is it to your advantage to switch?

The correct answer is ‘Yes’. But why?

I’ll give you the standard explanation, then I’ll give you an obvious explanation.

The standard explanation goes like this:

There is a 1/3 chance your first choice is the correct door. There is a 2/3 chance it is not the correct door. Since Monty always opens a door with a goat behind it, you are certain to pick the car 2/3’s of the time if you switch.

I’m not sure if this image helps, but maybe it does. The next one will.

A lot of people find that explanation challenging. So let me make it easy for you with a more obvious version of this problem.

Monty Hall Problem 2.0

Imagine you’re on a TV game show, and you’re given the choice of 100 doors: Behind one door is a car; behind the other 99 doors are goats. You don’t know which is the correct door. Suppose you pick Door Number 1. Now the host, Monty, who knows what’s behind the other doors, always opens 98 doors to reveal goats behind each one, say Door Numbers 2 through 99. He then says, “Before opening door Number 1, I’m offering you the chance to switch to Door Number 100?” Is it to your advantage to switch?

The correct answer, and this is hopefully more obvious this time, is ‘Yes.’

You have a 1/100 probability of picking the correct door with your first choice. That means there is a 99/100 probability the car is behind another one of the 99 doors. The fact that Monty opens some of those doors doesn’t change that initial probability. You already knew that 98 of the other doors have goats!

So after choosing that initial door, suppose you could then switch to the 99 other doors and be certain that if the car was there, you would get it. By opening the 98 other doors with goats, Monty is guaranteeing that you get the car if it is not behind Door Number 1.

Evolving author of “Does my algorithm have a mental health problem?”, “Why do we die?” and “The dark side of information proliferation.”

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