Thanks Steve, that way of explaining it seems to help. So, the events are interdependent because he can't open the door that you selected. I think it is making more sense now why it is advantageous to switch your choice.
I figured out from my own combinations why switching choices yields a 2/3 advantage: In the games where the combinatorically expected choice pattern is not allowed, the games are not discarded ("The show must go on."); so, instead, such games are played pursuant to the only remaining choice pattern allowed; and, that results in two more wins per six games played, yielding an average of 4 wins per 6 games, a 2/3 advantage.
I don't see in your summary that you are using the correct model that the man who reveals the goats behind unchosen doors has actual information about where the goats are. So there are 9 equal possibilities of where the prize is and what the first chosen door is. Sometimes the host has a choice of two doors and sometimes the choice is forced to one remaining door, but this does not affect the way the total probability of 1 is divided between the 9 choices of prize and first chosen door. Case I ) With probability 3/9=1/3, the chosen door is where the prize is. The host opens the door behind which is goat A or goat B, so Strategy0 (don't switch) wins in this case and Strategy B (always switch to the unchosen door) loses in this case. Case II ) With probability 6/9=2/3, the chosen door is not where the prize is. The host opens the door behind which is goat A or goat B. The other goat is behind the chosen door. So Strategy0 (don't switch) loses in this case and Strategy B (always switch to the unchosen door) wins in this case.
I woke up earlier than usual this morning, and a simple explanation the Monty Hall problem dawned on me: You have a 2/3 chance that your first choice will be a goat door; and, if it is a goat door, you will win by switching after Monty Hall eliminates the other goat door. Said another way, you only have a 1/3 chance of losing by first choosing the grand prize door, having one goat door opened and eliminated, and then switching to the other goat door.
Isn’t math cool? You can even apply it to choosing a wife. I switched doors and now I have a boat. Please Register or Log in to view the hidden image! Optimizing Your Wife Monty Hall: Try It-Play the Game Great Explanation
Beware the "it's so simple" solution: you've thought about this for a long time. Even though the solution now appears so simple and obvious to you, it's likely that someone else who is struggling to understand it will not "get it" from your explanation. But good for you, though. I'm glad that somebody can change their mind on this forum...
