The Monty Hall Problem

The Monty Hall problem is a famous paradoxical brain teaser and is based on the American television game show Let’s Make a Deal and named after its original host, Monty Hall.

It was first posed by the statistician Steve Selvin and was popularised by Marilyn vos Savant, a Parade magazine columnist famous for once having the highest IQ in the world.

This problem became so prevalent because the answer seemed to be rather counterintuitive and revealed how humans are simply not wired to understand probability and statistics that easily.

So what’s the problem?

Imagine you are on a game show and in front of you are three doors. You’re told that one of the doors is hiding a prize and then asked to pick one at random. Once you’ve picked a door, Monty (our lovely game show host) will open one of the other two doors to show it has nothing behind it. You are then asked whether you want to switch doors? What do you do?

Important Assumptions:

1. All doors are identical in appearance

2. The prize is equally likely to be behind any door

3. Out of the two remaining doors, Monty always opens a door which has nothing behind it

So if you took a quick ‘intuitive’ approach to this problem, it seems like once Monty has revealed that one of the doors has nothing behind it, you essentially have a 50–50 chance of picking the winning door meaning it doesn’t really matter whether you swap or not.

The problem with this logic is that Monty’s action cannot be considered a random, independent event because he knows where the prize is hidden and therefore is constrained to choosing a door which is hiding nothing behind it.

Let’s look at the chance of winning if our strategy is to swap doors. If we initially select the winning door (probability of 1/3), then swapping will certainly result in a loss where as if we initially select an empty door (probability of 2/3), then swapping will certainly result in a win.

Therefore, the chance of finally selecting the winning door is 2/3 (instead of 50% as previously mentioned) if we always choose to swap.

Weird isn’t it?

We can look at it another way and analyse the nine different combinations of our first selection and winning door.

As with any probability you calculate, it only truly holds when you have infinite rolls or selections. So if you were to play the game 1000s of times with the swapping strategy, you will find that you consistently win much more often than not.

So if you every happen to be a contestant on this type of gameshow, just remember to swap doors and hope probability is on your side.