The Time Everyone “Corrected” the World’s Smartest Woman (2015)

That he always opens a door that is not the contestant's choice is a necessary implication of saying that he opens "another" door. That he opens a door containing a goat is similarly implicated. It's not even an implication, really; it's just definitional in vernacular English. And in any event, any of the academics writing in would have been perfectly familiar with the semantic structure of such logic puzzles.

Also, most of the population of the country would have been familiar with the game show. Let's Make a Deal was one of the most popular programs on television. Even if they didn't particularly like it, people didn't have much of a TV program selection back then. And even if someone didn't regularly watch TV, it's likely they would have seen it and been familiar w/ the rules regardless simply because it was so widely watched around them. The vehemence of the respondents may even have been an effect of their familiarity, unable to recognize and accept that they'd missed a crucial analytical element after having watched the show so many times before.

What makes the puzzle counterintuitive, IMO, is that you're actually answering two different questions: the first before the host makes his selection, and the second (the only one that matters) after he makes his selection. Developing the habit of rigorously "updating your priors"--that is, recognizing when the available evidence has changed, requiring a reassessment--is an applied skill that even experts aren't particularly good at. (Nor was I the first time I read the problem.) The real-life narrative structure actually makes for a great logic puzzle as people are primed to miss the final constraint (his selection) for what it is. They're used to analyzing static situations, or a series of linked static situations--solve X correctly to solve Y. That makes the Monty Hall Problem peculiarly distinct from the typical type of logic puzzle that simply includes a misleading, irrelevant constraint or requires resolving a complex set of constraints.

There are still N possible outcomes, and you have 2 of them instead of 1. Your chance of winning is 2/N instead of 1/N and has exactly doubled.

And most lottery games allow numbers to be re-used, so you haven't "eliminated" anything. If you exhaustively bought all N number combinations, you have a 100% chance of winning, but you also have a decent chance to split the pot with someone else who also bought the winning numbers.

Next time your family screams and shouts at you that you're wrong, maybe you should actually listen to what they're saying?

I still don't think I really "get" it, but this explanation makes the most sense to me. Thanks!

Like of course I would switch and go for the coin toss, but I don't know how to convert that certainty into actually probabilities, tho I'm sure that the odds of my choices are not 1/3 versus 1/2, because there's probably some interplay.

If I think about it more I start to get confused again, along the lines of: does the first choice even matter then? and what if I reverse my first and second choices (as in I pick 1, then switch to 3, but what if I picked 3 then switched to 1), how can I still have better odds by switching, since there's only 1 right answer? But sticking with the coin toss, I feel, "OK this makes sense."

The 2 goats didn't make sense to me but the host perspective does I think, like:

after player's first choice, there's two cases.

Either player picked the car, or they didn't.

If player picked the car, then host has two choices to pick from, if player didn't pick the car, the host has 1 choice to pick from. (At this point you can sort of imagine a binary tree with 3 leaves and 5 nodes)

So either player picked the car the first time with a 1/3 chance (so, less likely), or didn't (more likely).

So it's more likely for player to miss the first time (unless you're a precog psychic like me), which, "paradoxically (but not really)", by missing (by failing), player actually improves odds for second guess. So, on average the second guess is more likely to hit, because, on average, the first guess is more likely to miss.

Now finally it makes sense to me. Because it's more likely to miss on first guess, it's also more likely to hit on second guess, so it's more profitable to switch, on average.

Do I get it now? I think I do! Woot! :P :) xx

You’re ignoring the fact that you already have information when it switches to two doors.