"Anyways, I'd now like a guess at a further parameter I can't easily calculate from first principles - the chance that a wizard ever makes it to fourth-circle, depending on whether they started out INT 15, INT 16, or INT 17.  Meritxell or anyone?"

Meritxell does not know and is very annoyed about it.

"I'm not totally sure there are good statistics on that - the most famous wizards tend to have really high intelligence, though, even though in principle you'd think a fifteen with an expensive headband would have the same odds as a native 19 would..."

"Not once the native 19 got a +2 headband, maybe.  Also remind me at some point to explain my theory that Intelligence is only one piece of what makes a really smart person and native 19s probably have more of the other pieces than a 15 with a +4 headband."

"If you can give me any random numerical facts you know that seem like they might possibly be related to how Intelligence affects wizard career attainment, I can see whether those random facts pin down the probable truth about the thing I want to know, given my background knowledge about how things likely fit together?"

"People say that Nefreti Clepati is a native 21, but they might be making that up."

"People say that Aroden was at 35 before he ascended, having invented a dozen new kinds of enhancement more complicated than the existing ones to go past what was even understood to be possible."

"The first student in our year to make second circle was Aspex Leron, and he's an 18."

"You meet more native 16s at the Worldwound than native 15s or native 17s, and that's typically fifth circle casters or higher. The average age to third circle is ten years, and it's known to be faster, but not lots faster - maybe six or eight years - if you're smarter, though also very smart people sometimes get stuck and obviously this is conditional on you being at the Worldwound and using magic where it matters. 

The youngest fourth-circle wizard in Cheliax is a native 18."

"They're plausibly not making it up about Nefreti, there should be approximately one native INT 21 in all of Golarion."

"I'm getting the impression Aroden was kind of a cool guy.  Just to check, was he by any chance hinted to be from some mysterious other world?  Because a lot of things I'm hearing about him sound like things a dath ilani would try."

Keltham is writing something on the white-wall while he asks this.

Is one allowed to say nice things about Aroden. Who even knows anymore.

"I've never heard that but he was mortal eight thousand years ago, there's a lot that isn't remembered."

"The ancient Azlanti might've been partway to Civilization when they were destroyed. They're said to have invented a lot that isn't remembered."

"How long did it take him to become a god?  If it was more than a couple of hundred years, he wasn't secretly a dath ilani, there's no way it'll take me that long if I can do it at all."

Nervous giggles. 

"He spent thousands of years mortal before he raised the Starstone from the sea, set the protections around it, and ascended," says Meritxell.

"Probably not one of my fellow flying-machine passengers then.  That sounds more like how long you'd take if you weren't starting with Civilization's knowledge and had to work out everything yourself the hard way, which, to be clear, is overwhelmingly more respectable than any career path I'd ever even consider."

"Carissa, more INT 15s or INT 17s among 5th-circles at the Worldwound?"

"I'm not sure because the seventeens are more likely to mention it but I think more seventeens."

Keltham finishes writing:

INT 15s:     INT 16s:     INT 17s:
   150             50              10

5th◁15      5th◁16       5th◁17
  2/100        10/100        40/100

"Suppose that among those who train to be wizards at all, there's a hundred-fifty INT 15s for every fifty INT 16s and every ten INT 17s."

"Suppose that 2% of the INT 15s, 10% of the INT 16s, and 40% of the INT 17s, become fifth-circle wizards, and that all kinds of fifth-circle wizard are equally represented at the Worldwound."

"What's the relative chance that a fifth-circle you meet at the Worldwound has native INT 15 versus native INT 17, if you know it's one of the two?  Raise an open hand if you think you've got it, closed hand if you think you're not going to get it."

3 of the int 15s make it, 5 of the int 16s, 4 of the int 17s. 

They all get it pretty quickly.

"A valid deduction, but not exactly the answer to the exact question I asked.  What's the actual chance that they're an INT 17, given that it's that or INT 15?"

That barely even makes sense to think of as a separate problem? 7 wizards who are one or the other; 4 are int 17 and 3 are int 15, so 3/7th and 4/7th respectively. 

Paying attention to the exact question has sometimes been known to count for something in more complicated problems like these, just saying.

Suppose you didn't know that somebody from your class was going to make fifth-circle wizard, if they were just a random person from your class, what would be the chance they were an INT 15 vs. INT 17?

15:1.

What's that in chances out of 100 that they're INT 17?

Around 6%.

"Well, suppose we were trying to solve an old murder mystery, Death At Ostenso Academy, which happened forty years back.  We've got a piece of evidence that their INT was between 15 and 17.  Another piece of evidence rules out all the INT 16s because they were all taking a specialized class at the time.  Going on 'priors', I think maybe priors would be the best Taldane translation, we'd say there was only a 6% chance of the murderer being an INT 17, because most students at Ostenso Academy aren't INT 17s."

"However, now suppose we get a new piece of evidence about a later murder, clearly connected to the old one, which appears to have been committed by a fifth-circle wizard who served at the Worldwound.  Suppose, somebody says, that both murders were committed by the same person.  In that case, we now think that the previous murder was likely committed by an INT 17 student, with probability of 57% for that and 43% for INT 15."

They follow the logic, or are again hiding that they don't.

"It's been - difficult for me to present this properly, I think, because I'm new to Golarion and don't know what problems would make a good point of it - but there's a way of thinking that sees everything in probabilities that shift as you learn new things, because of how facts are entangled with other facts.  Knowing that somebody made fifth-circle doesn't tell us their Intelligence for certain, but at INT 17 it's four times as likely than at INT 16 and twenty times as likely than INT 15, at least based on these numbers I made up - though not ungroundedly so, since even these made-up numbers were constrained by math I knew for the probable shape of the general population curves for how many INT 17s vs INT 15s, combined with Carissa's observation that there's more INT 16s than 15s or 17s among Worldwound 5th-circles, and slightly more 17s than 15s."

"If you only take into account the facts that overwhelmingly determine an answer, you'll miss an awful lot of facts and observations that shift probabilities a noticeable amount even if they don't shift them to ninety-nine hundredths."

"If you consider the original murder mystery I tried to offer you, what a dath ilani kid would've known to do - based on the way their parents talk, or some implicit aspect of previous training that I'm too young to remember and see the implications of - would be to ask about the probability that Carissa could've summoned a lantern archon, compared to Keltham being known to be able to do so.  Or how likely it is that, if you imagine the world where Carissa is the murderer, that she got into a heated argument with Meritxell about something the previous day, compared to the world where there was nothing to murder Meritxell about."

"We might say that the priors make Keltham and Carissa equally likely so far as we know, we guess that Carissa is half as likely to have Summon Monster III in her spellbook compared to Keltham's certainty of being able to pray for it, and that Carissa is four times as likely to have something to argue about with Meritxell if we imagine ourselves in the world where she had some reason to kill Meritxell compared to the world where Keltham had some reason to kill Meritxell.  Then what's the chance that Carissa, versus Keltham, killed Meritxell?  So far as we know."

"...two thirds? From saying that it's twice as likely to be Carissa as Keltham, which is what those numbers multiplied out to."

Not bad, though he should've remembered to ask for raised hands instead of just the answer.

"Yup.  Though it's important to remember that, in that case, the numbers really are ones I just made up.  Contrast to the way where I had a prior guess about the shape of the Intelligence distribution in the population from 14s to 18s, which matched up neatly with what Meritxell said about 15s vs 16s in the wizard academy, and then I asked for any relevant facts and Carissa had some notion of who you run into at the Worldwound.  My numbers were all compatible with those facts, which, if this were an actual mystery, might make them more able to support the weight of reasoning with them."

"Now a warning:  Until you've honed your ability to make up numbers and have them be constrained by other facts you know, you might be better off with your brain just feeling intuitively that some things make Carissa more or less likely to be the murderer, and not trying to know legibly to yourself what your brain is thinking.  If you make up a number like, Carissa is fifty percent likely to be able to cast Summon Monster III, and that number wasn't visibly constrained by any other facts you know, it's possible you might be better off by... rating it on a scale from 1-12 that you know doesn't actually mean anything, say, and letting your brain's intuitions take care of the rest."

"But this entire realm of thought is the realm that generalizes the notion of Validity that I told you about before - it's the realm of what is valid to say about uncertainty and things that might happen, which is most of what we ever want to think about.  The Law of that realm is the mathematics of probability."

"So to navigate an uncertain world Lawfully, for whatever the Law is worth to mortals there, you learn to cast Detect Probability and then Greater Make Up Probability and eventually end up with permanent Probability Sight."