> opening the door provides new information, but that information applies equally to both remaining doors.
This is intuitively obvious, but where it goes wrong.
You've chosen door A. Doors B and C remain, at least one hides a goat, and the host knows where everything is.
They can always open one of B and C, so the fact that they do so tells you nothing about A.
But if they open B it tells you they didn't open C, and vice versa. So when they open a door they are telling you nothing about the door you chose, and something about the other unopened door.
> ...what am I supposed to do?
You can try writing a simulation and watch it work, but another way to think about it is this.
Suppose you choose one door, put your hand on it, then close your eyes. Now you're given the choice of sticking, or switching to both the other doors. Clearly it's to your advantage to switch. Now with your eyes still closed the host opens one of the other doors. You can still stick, or switch. In effect, switching lets you choose both of the other doors, but you can ignore the one the host opened.
This is intuitively obvious, but where it goes wrong.
You've chosen door A. Doors B and C remain, at least one hides a goat, and the host knows where everything is.
They can always open one of B and C, so the fact that they do so tells you nothing about A.
But if they open B it tells you they didn't open C, and vice versa. So when they open a door they are telling you nothing about the door you chose, and something about the other unopened door.
> ...what am I supposed to do?
You can try writing a simulation and watch it work, but another way to think about it is this.
Suppose you choose one door, put your hand on it, then close your eyes. Now you're given the choice of sticking, or switching to both the other doors. Clearly it's to your advantage to switch. Now with your eyes still closed the host opens one of the other doors. You can still stick, or switch. In effect, switching lets you choose both of the other doors, but you can ignore the one the host opened.
Does that help?