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.

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