# Monty Hall Problem Explained

## Monty Hall Problem:

You must have seen some version of this in a TV game show:

The host shows you have three closed doors, with a car behind one of the doors and something invaluable behind the others. You get to pick a door, then the host opens one of the remaining doors and shows that it does not contain the car. Now, you have an option to switch the door from the one you picked initially to the one that the host left unopened.

Do you switch? Intuitively, it appears that the host didn’t divulge any information. It turns out that this intuition is not entirely correct. Lets use the new tool in our arsenal – graphical models to understand this –

Lets start by defining some variables:

D: The door with the car.

F: Your first choice.

H: The door opened by the host.

I: Is F = D?

D, F and H take values 1, 2 or 3 and I takes 0 or 1. D and 1 are unobserved, while F is observed. Until the host opens one of the doors, H is unobserved. Therefore, we get the following Bayesian network for our problem.

Note the directions of arrows – D and F are independent. I clearly depends on D and F and the doors picked by the host also depends on D and F. So, far you don’t know anything about D.