"Side question:  What are privacy customs and deliberate-incentive-structures around that info?  Does everyone know everyone else's Intelligence, or is the info concealed to force people to judge each other by their accomplishments which are thereby more incentivized?"

"Class rankings are posted. Students can say what they heard their Intelligence was, but at the age when you're in school it's not final, so you don't reliably know who's smartest and you do know who is top of the class," says Meritxell. It's her. She is the top of the class.

"Suppose I asked you to guess, if you don't know, if you do know that's obviously fine, what fraction of students in Ostenso wizard academy who graduate at all, will later be determined to have Intelligence 15, 16, or 17."

"More fifteens than sixteens and more sixteens than seventeens," says Meritxell. "Because you start with way more, and not that many wash out. Fifteen's plenty smart enough to be a good wizard."

"Numbers, Meritxell.  I'm not asking you to be perfectly right, I'm asking you to guess, based on what you've seen in your wizard academy that I haven't."

(Keltham has already estimated the ratios between +2.5sd, +3.0sd, and +3.5sd in terms of their improbability, and the ratio should be something like 15:5:1.)

"It'll be - ten fourteens, ten fifteens, a couple sixteens, maybe a seventeen?"

"What was your total class size like?"

"A little over two hundred, in Ostenso. Westcrown's bigger, I think."

"Corentyn's also bigger."

"And the population ratios should be - I've been unwisely trying to work this out inside my head, because it's more impressive that way, even if I have a higher chance of screwing up instead* - one INT 18 to six or seven INT 17s to forty INT 16s to two hundred INT 15s to six hundred fifty INT 14s.  So if you've got equal INT 14 and INT 15 representation, then any individual INT 15 is about three times as likely to get admitted to Ostenso wizard academy than an INT 14.  Does all that sound right?"

He'll write it down on the whitewall in case that helps:

INT 18:       1
INT 17:    6-7
INT 16:     40
INT 15:   200
INT 14:   650

(*)  In particular, Keltham is unthinkingly treating their Intelligence detector's integer output as a perfect floor() function instead of a noisy round() function, though it's not like he'd know that was wrong, and it gets him pretty close.

"That sounds about right?" Meritxell says uncertainly. "They admit the smartest."

"Wait," Asmodia says out loud.  "You just got to Golarion.  How is that something you know?"

"Oh?  That's an interesting question.  Why can't it just be another fragment of Law that I know and you don't?"

"By the way, Asmodia's not allowed to answer that question except by Message to me, somebody else has to try it instead."

"...that's how it is in dath ilan," says Gregoria, "except with the numbers shifted like everyone's got a headband on. Right?"

"Heh.  I suppose that's actually a simpler answer than the one I had in mind.  Why would those ratios be the same between dath ilan and Golarion, though?  What with us having a whole heritage-optimization program, and subsidies for kids expected to produce Civilization-approved positive externalities, and nutrition that doesn't vary much between kids."

"...for the same mysterious reason both worlds have humans at all?"

"Asmodia?"

"If you'd had those numbers memorized, you wouldn't have needed to work them out in your head and risk getting them wrong," Asmodia states.  "So it has to be math.  But it's the wrong kind of thing to be math, I can't figure out how to say it but - the part about Validity -"

"It's not necessarily true across all possible worlds.  You can coherently imagine a population or a wizard academy which looks different.  So, you conclude, it can't be a validity."

"What math did you do?  Or if you don't want to just tell me - can you show me what the calculations were, and then I can see if I can figure out from those what they mean?"

Asmodia could, without benefit of any Detect Thoughts, hear Meritxell thinking earlier about how she was top of the class in Ostenso, and Meritxell needs to be put back in her place.

"Actually, I did have some key results memorized, even though they were math results and not about the ratios in dath ilan per se.  If I had done calculations, they might have looked like - I didn't use this calculation exactly, but it's simpler and probably gives about the same answer -"

Definition of ^:

2^2 = 4
2^3 = 8
4^2.5 = 32

Definition of sqrt:

sqrt(9) = 3
sqrt(16) = 4
sqrt(x*x) = x
sqrt(2) = 1.41421

pi = 3.14159
e =  2.71828

let normal(x) = 1 / sqrt(2*pi)  *  e^( (-1/2) * (x^2) ) in

INT 18 ~ normal(4)
INT 17 ~ normal(3.5)
INT 16 ~ normal(3)
INT 15 ~ normal(2.5)
INT 14 ~ normal(2)

"I am not actually expecting you to figure this one out, Asmodia, it's several layers further into Probability and it would take multiple other concepts to understand what those numbers had to do with Intelligence in Golarion.  I'm only even writing it down on the white-wall because, I mean, when you asked me that way, I couldn't have lived with myself if I hadn't shown you the calculations."

"What's actually in there is something like - a fragment of Law that tells you a thing that is very likely true given only very few premises.  I could have used it to look at a room full of randomly selected people in Cheliax, none of whom were unusually tall, and then estimated from that how many very tall people of a given height there'd be in Cheliax."

The people trying to write an alter-Cheliax history book had better be praying to Nethys, because that's the only way that plan is possibly going to work.

This Carissa has actually encountered before, not in the standard curriculum for young wizards which doesn't much digress into things they don't absolutely need to know, but at the Worldwound, where wizards have a lot of long boring weeks between raids to discuss other things. It's part of how Intelligence is defined. 

The girls who haven't encountered it look very impressed, though. 

Note to self: talk to some mathematicians and see if other Keltham things are actually things Golarion - just not this specific group of students - is already familiar with.

"Actually, now that I think about it, the reason I know a key constant here is that the wizard who teleported me to Cheliax from the Worldwound seemed to know it?  In particular, he knew what I was talking about when I asked about the square root of the average squared difference from the average over Intelligence.  There's no reason he'd have known that for Intelligence unless this particular concept was already known in Golarion.  Which means that, in fact, somebody somewhere in Cheliax knows substantially more math about Probability than this class has ever heard of existing."

"I shall endeavor not to be distracted by trying to figure out how what why huh, and try to teach the basic layers of Probability from a dath ilani perspective instead.  But, somebody should actually let me know if I'm improvising something more poorly than existing teaching methods teach it.  You in particular, Isidre, I know you're reading the transcript on this."

Isidre??  Who's reading their transcripts named Isidre??