Introducing the problem

Imagine having three doors. One of which holds a sports car, and the other two, donkeys. Monty Hall is the host of this game show. He knows the contents that lie behind each door. Your goal here is to choose a door that you think holds the sports car. You make an initial choice. It's clear that the probability of your initial choice being right, is $\frac{1}{3}$. He opens another door for you, and it is known that the door he opens will always have a donkey behind it. Now, you may choose to either switch from your initial choice, or stay.

Well, math tells us that the probability of you getting the car if you switch now will be $\frac{2}{3}$. Hence, based on statistics, your choice should always be to switch.

Explanation

Let's discuss the intuition behind this.

We know that the probability that our initial choice holds the sports car is $\frac{1}{3}$. This never changes. However, once he opens a donkey door, he eliminates a possibility. Hence, the probability has to go somewhere; it has to concentrate towards the remaining door. That's why the remaining door has a higher chance of holding the sports car.

The intuition becomes much more obvious when we extend this scenario to a series of 100 doors. Our initial choice would carry a probability of $\frac{1}{100}$ of being the door with the sports car. Once he opens 98 donkey doors, only two doors remain: our initial choice, and the remaining closed door. In this scenario, it is clear that we should choose to switch, because our initial choice was made from a pool of 100 closed doors. As such, it is more likely that we made an incorrect initial choice, and that the remaining door holds the sports car.

Let's now summarise that idea:

• If we switch doors, we win only if we initially picked wrong ($\frac{2}{3}$)
• If we choose not to switch, we win only if we initially picked correctly ($\frac{1}{3}$)

Hence, we switch because we bet on the fact that we had a higher chance of picking the wrong door as compared to the right door in our initial choice.

The key insight is that Monty always opens a donkey door. This makes switching the more advantageous route.