#(hopefully without accidentally doxxing anyone)

[physicallyimprobable]

I’d be interested in hearing abt bass boosting!

Y'all gotta stop enabling me 'cause I'll never turn down an opportunity to exposit. So, this is in reference to the recent Monty Hall poll post, and my assertion that "my favorite explanation comes from just bass boosting the numbers involved". To bring us all onto the same page, let's first talk about the Monty Hall problem.

(Also, I tried a cute thing with coloring the words to make the argument easier to follow. I apologize if it's hard to read; you can copy-paste the text somewhere else and read it there if you mind.)

So the setup is, we start with a game. The game is usually ostensibly a part of the show Let's Make a Deal, hosted by Monty Hall, but this particular game never actually appeared on that show. I'll instead host it in a certain infinite casino (The Casgeno) which I often use for probability puzzles like this. Here's the Monty Hall game: You're given three sealed boxes. Two of them are empty, and one has some big desirable prize, like a couple thousand dollars or the keys to a new car or what have you. You are to guess which one of the boxes has the prize, and if you're right, you get to keep the prize.

Okay so there's pretty straightforwardly no strategy to this; you've got a one in three chance of victory. But here's the thing: After you make your guess, the host of the game (let's say his name is still Monty) opens one of the empty boxes and shows you there's nothing inside. Now there's two closed boxes in front of you, and you're offered one more choice: Do you stick with your previous guess, or do you switch to the other closed box?

So that's the usual setup. (Okay the usual one involves doors and goats, but the actual important bits are clearly the same.) Let's ignore this question for now and look at a different one, in which we bass boost the numbers:

This is a game we'll call the Monty World game. Instead of three boxes, Monty tells you that he's selected a person from your homeworld (the Casgeno overlaps infinitely many realities; don't worry about it) completely at random. Eight billion people, all equally likely to be selected. He tasks you with guessing the selected person; if you're correct then you get the big prize.

Again there is no strategy here. Let's say that you guess he randomly selected your skateboarding cousin Throckmorton as the person. But then, after you make this guess, Monty reveals more information. He tells you that the randomly-selected person is either your skateboarding cousin Throckmorton or Mohamed Badawy, a 24 year old retail worker from the Bûlâq al-Dakrûr district of Giza, Egypt. Again you're offered a choice: Do you stick to your cousin Throckmorton, or do you switch to Mohamed?

Unlike in the Monty Hall game, for the Monty World game the choice seems obvious. With eight billion people on the planet, it's vanishingly unlikely that the selected person was anyone you knew, much less your cousin Throckmorton specifically. Like, would Mohamed Badawy's name even come up if he wasn't the selected person? Switching to Mohamed Badawy, 24 year old retail worker from the Bûlâq al-Dakrûr district of Giza, takes your odds of winning from a statistical impossibility to a statistical guarantee:

The only way you lose by switching is if your cousin Throckmorton was, due to staggeringly miraculous bad luck, the actual selected person all along.

I like this modification because it is incredibly effective at pushing back against the "but there are only two options so it should be fifty-fifty" reflex. Yes, there are only two options, but they come from very different places: one of them was proposed by someone who knows nothing about the situation (you) and one of them was proposed by someone who knows everything about the situation (Monty).

Now let's get back to the boxes. If you think about it, the setup is basically the same, except for the scale of the numbers involved. If there were three people in the world instead of eight billion, then the new game and the old game would work exactly the same, beat for beat. The weirdness in the old game is specifically because there are exactly three options, the smallest possible number for which the game can work. When Monty opens the empty box, it just feels like he's revealing an empty box—that's where his hands are, after all—but what he's really doing is informing you that the prize is either in the box you guessed or in the remaining unopened box. (Notice that the revealed box itself is so unimportant that I haven't even given it a color.) The conclusion, then, is the same as before:

The only way you lose by switching is if your initial guess was, due to a quite reasonable one-in-three shot of bad luck, the actual nonempty box all along.

And, yeah, that's what managed to take the problem from "unintuitive but true" to "intuitive" to me. You can either stick with the option you proposed at random or switch to the option proposed by The Guy Who Knows How To Win. Obviously the second one is better, so switching improves your odds.