I played role of Monty with another person. They chose a door, I revealed a different door, and they switched (or not).

That got my head around it. Most of the time the door I "chose" to reveal was no choice at all. There was only one door I could open.

The best way to understand it:

Think of a version with 100 doors, 99 with goats and 1 with a car.

You choose a door, and the host opens 98 doors that have goats. Do you keep your randomly chosen door or switch to the single door that the host didn’t open?

I had it explained to me like this:

You pick one of three options, giving you a 1/3 chance of being correct, 2/3 odds you picked incorrectly. The host removes an option and give you the option to switch.

Your options then are -

Keep the same door: you win 1/3 of the time (your first guess was right)

Switch doors: you win 2/3 of the time (your first guess was wrong)

It really just comes down to, do I think I was right the first time, which was 1/3 odds, or wrong the first time, 2/3 odds.

Here's how I've explained it: Choose randomly between 3 doors. 1/3 of the time you end up with the door with the car, and switching loses. The other 2/3, you pick a door with a goat, the other door with the goat is eliminated, and switching wins.

Basically, P(lose when switching) = P(choosing correct door at first), and P(win when switching) = P(choosing any incorrect door at first).

Another way to verify the solution is to code a Monte Carlo simulation of Monty Hall

This can be done with ChatGPT. No code corrections needed.