aw3c2> Hm, then I do not understand it. "But you have been given information about the door he didn't open" was what made me comment.
aw3c2> If I chose a door before, then something happens that leads to only two doors being left, both those doors have the same probability so I could just choose the same door again.
The original said this:
CW> The information you have gained is not about the door you've chosen. It doesn't matter what door you choose, the host can always open a door to reveal a goat. So there is no information given about the door you've chosen, so the chances of that door containing the prize remain at 1/3.
CW> But you have been given information about the door he didn't open, because he didn't open it. That's why it's possible for the odds on that door of holding the prize can change.
So let's recap what's going on. There are three doors. For the sake of concreteness let's call them A, B, and C. You choose one of them. For the sake of concreteness let's suppose you choose A.
So now there are two doors, B and C, remaining unchosen by you. Currently those two doors, B and C, each have probability 1/3 of having the prize. The door you chose, door A, has probability 1/3 of holding the prize.
Now the host opens a door, taking care to open a door that does not hold the prize. So the pair {B,C} still has total probability of holding the prize, but you are being shown that one of them certainly does not. This doesn't affect the probability that your chosen door, door A, holds the prize -- the probability that the prize is behind door A is still 1/3.
The pair {B,C} still has total probability 2/3 of holding the prize. You're now given the choice of staying with A, or switching.
Quoting you again, you said:
aw3c2> something happens that leads to only two doors being left, both those doors have the same probability ...
That turns out not to be the case. Just because there are two doors they don't have to have equal probability of holding the prize, and in this case they don't. The probability that your door holds the prize has not changed and is still 1/3. The probability that the door neither chosen by you nor opened by the host holds the prize is now 2/3.
aw3c2> If I chose a door before, then something happens that leads to only two doors being left, both those doors have the same probability so I could just choose the same door again.
The original said this:
CW> The information you have gained is not about the door you've chosen. It doesn't matter what door you choose, the host can always open a door to reveal a goat. So there is no information given about the door you've chosen, so the chances of that door containing the prize remain at 1/3.
CW> But you have been given information about the door he didn't open, because he didn't open it. That's why it's possible for the odds on that door of holding the prize can change.
So let's recap what's going on. There are three doors. For the sake of concreteness let's call them A, B, and C. You choose one of them. For the sake of concreteness let's suppose you choose A.
So now there are two doors, B and C, remaining unchosen by you. Currently those two doors, B and C, each have probability 1/3 of having the prize. The door you chose, door A, has probability 1/3 of holding the prize.
Now the host opens a door, taking care to open a door that does not hold the prize. So the pair {B,C} still has total probability of holding the prize, but you are being shown that one of them certainly does not. This doesn't affect the probability that your chosen door, door A, holds the prize -- the probability that the prize is behind door A is still 1/3.
The pair {B,C} still has total probability 2/3 of holding the prize. You're now given the choice of staying with A, or switching.
Quoting you again, you said:
aw3c2> something happens that leads to only two doors being left, both those doors have the same probability ...
That turns out not to be the case. Just because there are two doors they don't have to have equal probability of holding the prize, and in this case they don't. The probability that your door holds the prize has not changed and is still 1/3. The probability that the door neither chosen by you nor opened by the host holds the prize is now 2/3.
Does that help?