The Monty Hall Problem

I came across this problem when I was quite young but I had never really understood it. Recently, I took on the challenge to comprehend this problem - and it’s actually quite simple. I will also give an intuitive example which should really explain the problem.

Introduction

Firstly, let’s understand what the Monty Hall problem is. Imagine there are three doors in front of you. Behind one of them is a valuable prize: a supercar. Behind the others, a trivial prize: a goat.

The twist is that you don’t know which door reveals the supercar and which reveals the goat. In this problem, you are asked to select a door of your choice (assume you select door 3). Next, a door is opened not randomly, but with 2 rules. (1) The door must not be the one you picked and (2) the door must have a goat behind it. Here, door 1 would open. Now, we know that door 1 has a goat behind it, but we don’t know what is behind doors 2 and 3. We are next given two options: we can either stick to door 3, or we can switch to door 2.

Stick or Switch?

If you are new to this problem (or don’t yet know how it works), and you haven’t done any calculations, you would presumably think that switching won’t make a difference: since there are 2 doors remaining, behind one a goat and the other a supercar, there should be a 50/50 chance of winning right? This is actually not right: one option gives an advantage and will make it more likely that you win yourself a brand new car. To understand this, let’s use some simple probabilities. When you are first asked to select a door, there’s a 33% chance that you pick the supercar, and a 67% chance that you do not. Thus, when we reveal a goat, the chances don’t transform to 50/50: the door of your choice still only has a 33% win rate. The other 2 doors have a combined win rate of 67%; so when we reveal a goat, we are left with the remaining door having a 67% win rate. This is why switching, on average, is the better option.

100 Doors

If you are still confused, don’t worry. Here’s an intuitive example that might aid your understanding: Imagine there are a 100 doors. 99 of them have goats behind them, while 1 has a car. Let’s say you randomly select door 73. There is a ridiculously low chance that you selected the right door (1%), and a high chance that another door has it (99%). Next, 98 doors behind which there is a goat open. There’s only 2 doors left. One which you picked, and one which most likely has a supercar behind it. If you think about this one, switching would give a massive boost to your likelihood of winning. Hopefully this helps you understand.

Simulation

If you’d like to visually see the problem, there are some good websites that fully simulate the Monty Hall problem. Feel free to check out this website - Note: If you click on the “Simulate” button, you can do 10-1000 trials instantly and see how many times you win when you switch vs when you stick.