So if I switch door and pick one at random it should be 50% right even if I pick the same one I had picked before.
I trust the smart people on this, but I'm going with the math goblins explanation from now on.
In the proper setup of the problem, at no point do you pick a door at random out of two.
You start with three doors.
* You choose one, and you have a 1/3 chance that you have picked the car.
* There is a 2/3 chance that the door is behind one of the other doors.
Now, and this is a key point, Monty knows where the goat is.
* Of the other two doors, he can choose one that hides a goat. That doesn't affect your chance of being right, which remains at 1/3.
* Of the other two doors, he does choose one that hides a goat. That doesn't affect your chance of being right, which remains at 1/3.
* Of the other two doors, he opens one that hides a goat. That doesn't affect your chance of being right, which remains at 1/3.
Now there is one other unopened door.
* The chance that the door you chose holds the car remains at 1/3;
* The chance the car is behind one of the other two doors is 2/3;
* One of those two doors has been opened!
So the other closed door has a 2/3 chance of holding the car.
The information you get is which door to open if you are allowed to switch.
So the argument "your first pick is always 1/3, so you should always switch" is wrong. That doesn't normally apply. It is true in this case, but not true in the general case of problems like this since probabilities change as new information is revealed.
It does reveal new information, just not about the door you chose.
> lets say that he opened two doors with goats instead, then what is the probability of A, ie your first guess being correct?
But you can't guarantee that he can do that. If you do guarantee that he can do that then you are giving me more information.
It's always the way, if you change the problem you change the analysis.
> So the argument "your first pick is always 1/3, so you should always switch" is wrong. That doesn't normally apply.
Just because it doesn't normally apply, that doesn't mean it's wrong in this case. It does apply in this case (as you say) and therefore it's right.
Knowing when it's right is hard, I agree.
Monty could have asked the question as "Do you want to stick with the door you chose or do you think it's in one of the other two?" and skipped even opening one of them but that wouldn't have been as fun to watch.