[tawan]

I disagree with your conclusion that the odds remain 2/3 even if the host doesn’t know where the prize is. If that were the case, it would mean that if the same scenario is repeated many many times, then in 2/3 of the cases you would still pick the door with the prize with the same strategy. But what about the instances where the host picks the door with prize before you even get a chance to pick the other door? These cases happen with a probability of 2/3 * 1/2 = 1/3. Therefore, you only get to choose in 2/3 of the cases and in 1/2 of those cases you have already picked the correct door the first time.

[8note]

Do you still have the opportunity to switch if he picks the winning door?

Even if you could, you already know that you're losing, unless he doesn't show you what's behind it.

If the game is still going, you know that he picked an empty door regardless of whether he knew he was doing it