The Monty Hall problem is a statistics puzzle from the 70’s based loosely on a game show that was aired on US telly back then.

I’m writing about it here as it makes you think about statistics and numbers in a way you might not have had needed to since you left school. Its not directly related to anything financial but it is good maths practise for your brain.

Imagine you are a contestant on a game show. The host shows you 3 doors and tells you that behind 2 of the doors is a goat, behind the 3rd door is your dream car.

You are asked to pick a door and told you get to keep whatever is behind it. At this stage in the game you have a 1/3 chance of getting the car, and 2/3 of getting a goat.

The host knows what is behind all 3 doors. After you have chosen your door he walks over and instead of opening your door he opens up another door which he knows contains a goat. He then gives you the option of sticking with your door or changing to the other unopened door. At this point there are 2 closed doors, one has the car, the other has a goat.

What do you do? and what are the odds of you correctly getting the car?

Monty hall door puzzle
Monty hall door puzzle

The Answer:

A lot of people incorrectly think that the odds of winning the car at this stage are 50/50.

the correct choice is to switch doors as it gives you a 2/3 chance of winning.┬áThe reason it isn’t a simple coin flip is that you already have previous information about the doors. The host opening a door he knows has a goat gives you information that helps you make a decision. Here’s the logic behind it:

When you first pick a door you have a 2/3 chance of getting a goat and 1/3 of getting the car. if you picked a goat door the host would open the other goat door meaning that switching would give you the 3rd door which has the car.

if you first picked the car door then the host would open any other door as both contain goats and a switch would give you the other goat.

As the odds of you picking a goat door first are 2/3 it means the odds of a switch move getting the car after the host has acted are also 2/3.

Note : There are various variations of this problem, the only reason this one is the most famous is due to the amount of responses that the problem got when it was published in Parade magazine, many people have trouble understanding why it isn’t a 50/50 choice.