Yeah. Monty "decides" to open a door. He's not "forced" to.

The correct thing to do in this case would maybe be to toss a coin, and switch randomly half of the potential cases. Then you go from getting the car 1/3 of the times to 1/2. You are using game theory to outsmart Monty.

except he says "how do you know he's not playing a trick on you?" etc. if he's trying to screw you over you're going from 100% to 0% by switching. you never get the option to switch if you were wrong in the first place. if you don't know if he's trying to screw you over, you can't assign exact probabilities to anything.

The worst thing to do must be to stubbornly hold on to your first choice no matter what. You can't be sure Monty is trying to screw you. Not good for the show.

Maybe people in general are mixing up "how do you know he's not going to screw you?" with "he's going to screw you?". Great difference.

You know you will get the car exactly on average 50% of the cases when he opens a door with a goat and offers a switch, if you pick any of two doors left (including the one you originally picked) randomly. For not being biased, flip a coin.

Of course, there may be a possibility there's no car there at all, but let's assume he opens all the doors after the choices and shows it. Otherwise he in fact could screw you, offering you a switch only when there is a car there at all and you happened to pick it.