The concept of being so Lawful that you can figure out Laws authority hasn't told you about, is one that Pilar is still struggling with.  There is little she will not do for her Lord Asmodeus, however, and it has been made clear enough to her that this work is important.

There's also not many ways you can go down this path, if you only do the derivations that are allowed, and that, Pilar is good at.

P(no-book & Conspiracy) / P(Conspiracy)          P(no-book & Conspiracy)        P(Ordinary)
-------------------------------------------------------    =    ----------------------------------  *  --------------------
    P(no-book & Ordinary) / P(Ordinary)                 P(no-book & Ordinary)        P(Conspiracy)

She struggles momentarily what to do from there, but then she gets it.

P(no-book & Conspiracy) / P(Conspiracy)      P(Conspiracy)         P(no-book & Conspiracy)         P(Conspiracy ◁ no-book)
-------------------------------------------------------  *  --------------------   =   ----------------------------------   =   -----------------------------------
    P(no-book & Ordinary) / P(Ordinary)            P(Ordinary)              P(no-book & Ordinary)            P(Ordinary ◁ no-book)

"I have the answer in symbols, but I'm not quite sure what it means now that I have it," Pilar says.

He'll wait for others to catch up, and then explain.

The key quantity needed is P(Conspiracy) / P(Ordinary), or any other quantity which determines that one, which for this example means how likely Keltham thought Conspiracy versus Ordinary was before the absent-book observation.  Say, if Keltham previously thought a Conspiracy was 10% as likely as the Ordinary world, and then he saw the no-book observation, which in Keltham's estimate was ~2.5 times as likely if there's a Conspiracy than in the Ordinary world, the result is that a Conspiracy then becomes 25% as likely as the Ordinary world.  So it goes from 'a tenth as likely' to 'a quarter as likely'.

Meritxell's observation is that Keltham also ought to take into account that Ione told him about her book powers in the first place, even though that makes it harder for Governance to control Keltham's information and potentially makes him suspicious if they try to.  Let's say that's 1/20 as likely in the Conspiracy world than in the Ordinary one - not impossible, there's weird side possibilities where that could happen because Governance wasn't on the ball there, but yeah, sure, unlikely.  So if Keltham took that argument at face value, he'd then multiply 'Conspiracy a quarter as likely as Ordinary' by 'observation a twentieth as likely on Conspiracy as Ordinary' and get 'an eightieth as likely'.

But then of course we have to consider how the three most obvious interesting-background girls, Carissa, Ione, and Pilar, all disappeared at lunch today, supposedly to copy Invisibility spells.  Since Invisibility has nothing to do with interesting backgrounds, the probability of randomly getting that set of three women selected from twelve possibilities is 12 * 11 * 10 / 3 * 2 * 1, or 220:1.  So the new odds of Conspiracy over Ordinary are 'an eightieth as likely' times '220 times as likely' or a bit less than 'three times as likely', 2.75 times as likely to be exact.

Keltham should therefore now consider himself 2.75 times as likely to be in a Dark Governance Conspiracy world than an Ordinary world.  Does that reasoning sound correct to everyone?

.....no? 

"Oh, and this should hopefully be obvious from lesson context, but if you know an actual real story that totally refutes that whole argument - Ione, Carissa, Pilar, or anyone else who happens to know for sure - don't just blurt it out yet."

She has a lie prepared but fine, then. "Okay, I won't explain, but it seems like it's some kind of error for that to be two hundred and twenty times likelier if - oh, wait, I can formalize that, there are possible nonrandom explanations for the event which aren't the specific theory you have."

"Carissa, even if you don't just tell them whatever the exact story is, you're not allowed to tell them the abstract form of the answer generalized from your knowledge of whatever the exact story is.  You, Ione, and Pilar are all sitting this one out."

"Asmodia, Meritxell, you're allowed to answer too, this time, after a one-minute pause.  Because an obvious avenue for this challenge, in advance of formalizing anything, is to try to say informally what's wrong with my reasoning, and then formalize it.  My guess is that you two don't have as much of an advantage at informal argument, so it's safe to let you out of the Holding Cell temporarily; I could be wrong."

"Carissa has an interesting backstory?" says Gregoria.

"Apparently not on the same level as Ione and Pilar by Golarion standards, but if you look at it from my perspective, randomly landing next to an INT 18 third-circle, with better spellcraft than fifth-circles with intelligence headbands, who can scaffold to spellsilver six feet away etcetera, is still a pretty interesting background even leaving out some other things."

"Though there's also the seed of a stronger counterargument there, if you want to double down on it and think you can translate it more into the language of probability."

"Any wizard who's the staff wizard for their unit at the Worldwound is going to be really good at something," Gregoria says. "If you're walking down the street and meet someone that cool, something weird happened; if you wander into a random fortress at the Worldwound it's not. ....I don't know if that's in the direction of the stronger counterargument. It just feels like you're double-counting or something. And if the group had been different people, and later you'd discovered, I don't know, that I'm the bastard daughter of the Baron of Arenys, then you could say that was proof of a conspiracy too."

"I agree that's something to be wary of in general - you don't want to look at what happened, and afterwards draw an exact category around it.  In this case, though, I had in fact formed Carissa, Ione, Pilar, and Possibly Asmodia as a large distinguished mental category, in advance of seeing Carissa Ione and Pilar vanish mysteriously at lunch.  It wasn't at all drawn up afterwards, in this particular case."

"Not to completely throw your fine argument out the window, however, let's say that the Conspiracy hypothesis really allowed for two possibilities:  First, that Carissa-Ione-Pilar would vanish at lunch, and second, that Carissa-Ione-Pilar-Asmodia would vanish at lunch.  So we now have the new probabilities," Keltham writes some more.

P(Carissa-Ione-Pilar ◁ Conspiracy) = 1/2
P(Carissa-Ione-Pilar-Asmodia ◁ Conspiracy) = 1/2
P(Carissa-Ione-Pilar ◁ Ordinary) = 1/220
     =>
P(Conspiracy ◁ Carissa-Ione-Pilar) / P(Ordinary ◁ Carissa-Ione-Pilar) = 1/80 * (1/2 / 1/220) = 1/80 * 110 = 1.375

"So in the light of your new argument, I shall concede that the Nefarious Governance Conspiracy is only half as relatively likely as I previously calculated, 11/8ths as likely as an Ordinary world, not 22/8ths as likely."

"But this was surely the only error in my calculations and my totally Lawful argument; I doubt you can find any others."

Part of Keltham's brain also wants to start tracking the possibility that Gregoria is in fact the 'bastard??' daughter of the 'Baron??' of whatever, and Keltham tells it to shut up, either they'll end up dating or not.

"If they were a conspiracy, probably they wouldn't sneak off right in front of you very suspiciously," says Yaisa.

"Clearly that's exactly what they want me to think."

Yaisa has absolutely no idea what to make of that.

Asmodia's minute is up, now.  And during that minute, she's also thought about what her - Sponsor? - probably wants, from this whole situation.  And while Asmodia's conclusion is mostly that she doesn't know, Ione did foretell the Zon-Kuthon attack and Pilar threw herself in front of Keltham.  That's some evidence - Asmodia's not trying to Probabilize it right now - that her Sponsor does prefer that Project Lawful continue, rather than shutting down.  So she's not hindering her Sponsor's work, probably, if she points out flaws in Keltham's argument here.

"Even if that is what they want you to think, you can't reasonably say that the Conspiracy premise predicts with total 100% probability that some set of girls will mysteriously disappear at lunch," Asmodia says.  "There should be a third possibility, that nothing happens."

"But that possibility didn't happen, so I can eliminate it from my calculations now."

"If that were true, you could eliminate the Carissa-Ione-Pilar-Asmodia possibility, since it didn't happen, and leave Conspiracy putting 100% on Carissa-Ione-Pilar again, and then Ordinary would also put 100% on that since everything else didn't happen."

"I'm confused.  Why can't I just say that Conspiracy gets to allocate 100% of its probability to whatever happened, but Ordinary doesn't?"

"I'm not sure what is going to be Lawful but I can very strongly guess that will not be what Lawfulness looks like."

"With ninety-nine point nine nine percent probability."

"There's a saying out of dath ilan, don't criticize people for using what you think are the wrong general principles for arriving to their correct answer; if you're right that they're using the wrong general principles, you can wait for an occasion when they're wrong, and point out the error then."

"So I'm not going to criticize you for claiming you could make ten thousand statements about that strong, as confident as you are now, and be wrong on average about once."

"I'll wait until you're actually wrong.  Which will take, say, somewhere about four more occasions.  Fewer if I actively try to lure you into it."

"I await your lures."

If she never flirts with Keltham, Sevar will know the tropes aren't real.

Meritxell feels that Asmodia doesn't even want Keltham and is just flirting with him to mess with Meritxell, and Meritxell is going to destroy her for it. Somehow.

A watching Security thinks Meritxell could stand to learn a valuable life lesson about the importance of visibly flirting literally at all.

Well someone obviously came to a fast decision about whether she wants to be his nearly-ace harem member.  Or maybe, only about whether she might want to be; but sufficient to start flirting about it, at any rate.

"But yes, you aren't particularly allowed to reallocate the probability of observations after you observe them.  One way of thinking about it:  Probability of Carissa-Ione-Pilar given Conspiracy is meant to capture the probability that Keltham would assign if you asked him beforehand about what was likely to happen at lunch.  And all of the possible mutually-exclusive observations you can consider explicitly, need probabilities summing to at most 1.  The probability that exactly Carissa-Ione-Pilar go off at lunch, that Carissa-Ione-Pilar-Asmodia go off at lunch, and that nobody goes off at lunch, must sum to at most 1.  Realistically less than 1 because, say, I should've assigned some probability that, for example, just Ione and Pilar would run off, or that the whole lunch would be disrupted by a Nidal attack."

"This is why, especially at your stage of learning, we'd consider it a much more believable probability estimate, if you say in advance what will probably happen, compared to if you look back afterwards and come up with a reason why that event was clearly very predictable."

"If I'd thought in advance about the probabilities that Ostenso wizard academy would have any available books listing cleric spells, I could have considered the possibility that it would, and that it wouldn't, in an Ordinary world, and then considered the probability that Governance would need to stall me, in a Conspiracy world, versus having appropriately doctored books already made up.  If I'd done that in advance, I couldn't be influenced, either when thinking about the Ordinary world, or the Conspiracy world, by wanting to make a world give the 'correct' result, because I wouldn't know, in advance, which result I would observe."

"Since I didn't think that quickly, I had to go back and make up the probabilities afterwards.  And then there's a risk that, for example, maybe you don't want to believe in the Conspiracy world, so once you know that the real result is no-book, you're tempted to twist things around inside your mind and come up with a story for how the Conspiracy world would definitely have finished making up a doctored book of cleric spells by then, because they would anticipate my question and not want me to be suspicious about an absent book."

"If I make my prediction in advance, my mind will be less tempted to do that because I won't know that an absent book is the particular outcome that the Conspiracy world needs to dispredict in order for me to end up not believing that unpleasant thing."

"Dath ilani do have any skills for fighting that, for being able to come up with reasonable probabilities even after the fact; but those skills are difficult even for average dath ilani, which you frankly are not at this point.  You can ask two groups of medium-rank Keepers for their conditional probabilities, one group before and one group after they find out the real answer on some problem, and there'll be no significant systematic difference between the groups.  Because Keepers, that's why.  You would not find zero detectable difference between groups of dath ilani with around my age and intelligence levels, asked to say the likelihood that there'd be no book of cleric spells available, in the Conspiracy world, and in the Ordinary world, both before and after they actually got that result."

"Everyone gets trained in skills that partially protect against assigning-different-conditional-probabilities-to-outcomes-once-you-know-what-the-outcomes-are, but that's, like, diminishing the distortion by a factor of five, not driving the distortion down to undetectable levels.  That's the realm of, I would expect, sufficiently old devils, or gods - but definitely, it is known, Keepers rank four and up."

"Because - I would assume, and among other reasons - they practice really really hard until they stop doing it wrong, and not everybody has the time for that."

"So even I try to make my predictions in advance, if I remember and I'm not too lazy and it's important.  So you at this point should strongly, though not invincibly, question and distrust any probabilities for observations that you make up after seeing the answer."

"What's the screening like to become a Keeper," says Tonia, "are you allowed to know?"

"Be smarter than us, learn faster than us, be better at this particular stuff than us, be twenty times more naturally Lawful than I am, and my guess is that there are at the very least different tracks for Keepers who don't lean pretty heavily Good."