The Monty-Hall Problem — A Tale of Two Groups
The Monty-Hall problem is an interesting example of the importance of context in statistics. In this post I will explain the idea behind this mind boggling problem and give a very brief explanation of why it matters now more than ever. For those of you not familiar with the problem, the Monty-Hall problem goes as follows:
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say №1, and the host, who knows what’s behind the doors, opens another door, say №3, which has a goat. He then says to you, “Do you want to pick door №2?” Is it to your advantage to switch your choice?
Quoted from https://en.wikipedia.org/wiki/Monty_Hall_problem
If you say no, that there is now a 50/50 chance that it could be behind either door, then your answer is neither unique nor is it correct. It infact doubles your chances to switch your choice, and here is why.
It’s simple. Once you pick a door, you have created a clear division and there are now no more individual doors, but groups of doors. In one group, there is the single door that you picked that has a 1/3rd chance of having the car behind it, giving that group a 1/3rd chance of having a car behind a door. In the second group there are two doors, each also with a 1/3rd chance of having the car behind them, giving the group a 2/3rds chance of having a car behind one of the doors.
Now, where most people go wrong is assuming that once the host opens one door to reveal a goat behind it, that the chance of the car being behind either of the two remaining doors is split equally between them, that is not true however, as the second group of doors is still guarenteed to have a 2/3rds chance of having a car behind a door within that group. As there is now only 1 door left within that group left unopened, that means that that door now has the entirety of the group’s 2/3rds chance that the car is behind it, double that of the first door that you picked.
And it is for that reason that it is always in your interest to switch doors.
