[mithr]

I think the reason it's so hard to understand is that we try to break the problem down into more digestible problems, and in this case, if you only consider the last stage of the game, it certainly seems like it should be 50:50 odds.

In other words, at the final round, you are presented with two doors. Ignore the fact that you already chose one, and assume you were starting at this point: you enter the game, see two doors, and need to pick the one you think hides a car. So intuitively, your odds of choosing which of two doors has the car are 1/2.

It's hard to really understand why it matters that the previous rounds occurred at all — why it matters how you narrowed the selection to these two doors. I don't think the 1000 door version makes that easier to understand, because the same thing is true there — if you come into a game, see two doors, and are asked to choose between them, it's very hard to understand why it matters whether in a previous round there were 2, 3, 10, or 1000 doors — there are only two now, when you make your choice.

[polygotdomain]

>It's hard to really understand why it matters that the previous rounds occurred at all

It's not hard at all. Humans are not perfectly rational beings. It's not purely about odds, it's about emotions and psychology. In a version where there's no previous selection and it's 50:50, than I pick one and live with it. In the canonical example, the previous selection means that the contestant has already staked their claim, and changing it to the loosing door would have a different emotional response than a simple 50:50 shot with no previous selection. There's a reason why Roulette shows you the last X spins and whether they're odd/even, red/black... because humans make the assumption that the last disconnected data point somehow impacts the current one.

You're also glossing over the fact that there was a 1:3 chance you picked the right door to begin with, and 2:3 chance the "other" door was right. The last round isn't 50:50, as you so claim, because there was prior information

This is the same reason why the 1000 door example helps explain things; because the math is fundamentally the same, yet significantly more imbalanced. We can also think about how we'd feel about our selection in the 1000 door version, which is likely significantly less confidence, and therefore more likelihood of switching. Whether it's 3 doors, or 1000, the math still say switching is optimal, and our psychology and emotions deal with the choice of switching different in each case.

[rspeele]

That's especially tempting since we're used to probability problems often being set up where considering previous state is the sucker's answer. E.g. a fair coin has landed on heads three times in a row, what are the odds of it being tails on the next throw?