[dghf]

How does that contradict what I said? The way the game is set up, one of your choices -- stick or switch -- is guaranteed to win.

Your original door will be correct 1/3 of the time and wrong 2/3 of the time.

Therefore switching will be the winning move 2/3 of the time.

[saghm]

Your explanation isn't wrong, but it's never quite resonated with me because it feels almost like a magic trick than something that follows intuition. Like seeing a magician perform a trick, it doesn't quite convey to me the "why" as much as the "what", and even though I know there's no actual magic, I still feel like I'm left having to figure out what happened on my own.

The idea that finally made it click for me is that Monty has to choose one of the doors to open, and because he knows which door has which thing behind it, he'll never pick the door with the winning prize. That means the fact that he didn't pick the other door is potentially meaningful; unless I picked the right door on my first try, it's guaranteed to be the one he didn't open, because he never opens the right door on his own. His choice communicates meaningful information to me because it's not random, and that part while seemingly obvious gets left implicit in almost every attempt to explain this that I've seen.

Another intuitive way to explain it would be to imagine that the step of opening one door is removed, and instead you're given the option of either sticking with your original door or swapping to all of the other doors and winning if it's any of them. It's much more obvious that it would be a better strategy to swap, and then if you add back the step where he happens to open all of the other doors that aren't what you picked or the right one, it shouldn't change the odds if you're picking all of the other doors. This clarifies why the 100 door case makes it an even better strategy to switch than the 3 door case; you're picking 99 doors and betting that it's behind one. The way people usually describe that formulation still often doesn't seem to explicitly talk about why the sleight of hand that opening 98 of the doors is a red herring though; people always seem to state it as if it's self-evident, and I feel like that misses the whole point of why this is unintuitive in the first place in favor of explaining in a way that clarifies little and only makes sense if you already understand in the first place.