Monty Hall Probability

This came up again. I also saw a great set of notes regarding Monty Hall, specifically that he has discretion, but I’m ignoring that. We’re going to pretend Monty Hall is perfectly honest and without discretion. He always opens a door to a goat

After some thinking about it, a better way to go is parse events by inputs and outcomes. For this purpose, I will rename the problem car, red goat, blue goat.

At the beginning, when you pick a door, you will get either a car, a red goat, or a blue goat.

Blue Goat
1/3 of the time, the blue goat will be behind the door you initially pick.

Monty Hall opens a door to the red goat. (remember, he has no discretion)

The other door has the car.

Thus in the blue goat scenario, you should always switch doors. Put another way, 1/3 of the time switching doors is correct. (Chance of a scenario occuring) times (possible outcomes of that scenario) so 1/3 chance switching is correct.

Red Goat
1/3 of the time, the red goat will be behind the door you initially pick. This scenario is largely identical to blue-goat above.

Monty Hall opens a door to the blue goat. (no discretion)

The other door has the car.

In the red goat scenario, you should always switch doors. Identically to above, an additional 1/3 of the time switching doors is correct.

This is identical to the blue goat scenario, so you can add these probabilities. Now we have 2/3 of the time, the contestant should switch doors.

Car

1/3 of the time, the car will be behind the door you initially pick.

Monty Hall opens a door to a goat.

Which door doesn’t matter, nor does which goat, because you already picked the car. In this situation, you should not switch; you should keep you initial guess.

By the same math, 1/3 of the time, you should not switch doors.

Total

Sanity check: Do these odds add to 1? Yes. 1/3 Blue Goat + 1/3 Red Goat + 1/3 Car = 1

Total probability is 1.

2/3 of the time, switching is correct.
1/3 of the time, switching is not correct.

It’s not a 50/50. Switching increases your odds of getting a car.

Again, assuming honest Monty, no discretion, you want the car more than the goat, etc.