"Though, I should also note, I might have to revisit that if it starts looking again like 'tropes' are going to be a thing, which, Carissa, remind me to tell you about this update later, I am increasingly convinced that they're just not."

"I am glad to hear it," says Carissa.

"Moving on!  I now introduce a new key definition, that of conditional probability:"

P(X◁Y)  =def  P(X & Y) / P(Y)

"For example, in the case of Int 15 wizards who make 5th-circle:"

P(INT 15) ~ 0.01
P(INT 15 & 5th-circle) ~ 0.0002
P(5th-circle ◁ INT 15) ~ 0.0002 / 0.01 = 0.02 = 2/100

"If we start with 100,000 Chelish citizens, there should be around 1000 of them with INT 15, and then 20 of those who become 5th-circle wizards according to the statistics I totally made up this morning, and so 2% of people become 5th-circle wizards conditional on them having INT 15."

"The symbol is easy to remember because you can imagine that, on the right side, it shows a wide pool of people with INT 15, diminishing on the left side to a narrower pool of people who have INT 15 and are 5th-circle wizards."

So far away that there is no distance and no time between here and there, Thellim is trying for the first time to look up how statistics work on her new home planet.

The conditionalization operator is written "P(X|Y)" using a neat, symmetrical, vertical bar.

It doesn't weigh much against all the other bad news she's seen in the last day, but it is still not great news about how sane statistical mathematics is liable to be in this place.

"So it's sort of like in logic, where you have a starting premise from which you can deduce further things according to rules, except instead of a starting premise it's a starting probability?" says Meritxell. 

"I'm not going to put you in the Holding Cell for that, but only because I didn't actually ask the advanced question you just went and answered."

"Yes.  Though to state it more exactly, the right-hand side of the operator represents a starting set of worlds, such that we consider the probability of the left-hand side's event only within those worlds.  We are not interested in the probability that a random Chelish citizen is a 5th-circle wizard; the total contribution there would be greater from INT 16s, or so it sounds like from what Carissa said.  We're not interested in the probability that a random Chelish citizen is an INT 15 and becomes a 5th-circle wizard.  We're interested in the probability that, if we select a random Chelish citizen, and then narrow our focus to only those worlds in which the random selection produced an INT 15, what is the chance within those worlds that the person becomes a 5th-circle wizard."

"Having assumed this fact away, it indeed becomes a premise for further deductions, as you say, just as if we were asking about whether it's valid that 'Y implies X' and for purposes of that validity are allowed to just assume Y is true.  While asking about the probability that somebody becomes a 5th-circle wizard, conditional on INT 15, we have available the assumed fact of their INT 15-ness."

Meritxell, who wants very badly to be in the Holding Cell, looks cheered about this.

Keltham has deduced as much, but he's not putting her in there until it reaches the point where he estimates she's answering questions too fast and disadvantaging other students from learning.  That's what it takes to get put in the Holding Cell!  And for so long as Meritxell hasn't forced him to toss her in by threatening the other students with spoilers, she'll just have to try harder in order to get in.

Anyways, as a basic comprehension check, how about if everyone invents and writes down a conditional probability, including the underlying P(Y) and P(X&Y).  First, in a case you're allowed to just make up.  Then, a realistic case, something where you know, or can constrainedly guess, the actual statistics.  No extreme stats where it's 100% or 0% of something.  Raise an open hand if you're done, closed hand if you think you can't do it.  (He's not expecting the latter, but why trust what you can cheaply verify.)

Pilar, who again has less practice than most Chelish citizens in her position with needing to constrain her own thought processes, gets it last; it takes her longer to work past her brain's repeated generation of forbidden suggestions that she can't write down for Keltham to check, like the probability that a citizen in Ostenso is a Lastwall spy.  Eventually Pilar does get a made-up example about the chance a 3rd-circle priest ever reaches 4th-circle, and a real example about the chance she scores in the top 10% of class given that she scores in the top two-thirds of the class.  She remembered and tracked this statistic because it determines who punishes who, but Keltham doesn't need to know why Pilar remembers it.

Somebody really should punish her for being slowest, but she can see about that later.

Quickly checking confirms that everybody got it basically right, modulo Yaisa who divided the number of girls who became wizards, by the number of wizards, to get the probability that a girl becomes a wizard, which... Yaisa maybe just needs to actually think about the meanings of the numbers, instead of writing symbols?  Keltham was probably ever trained to do that as a kid, but he doesn't know how he was trained, it's the sort of thing that gets buried implicitly into learning something else.  Maybe visualize the scaffolding between the numbers and reality like you could see it with Detect Magic?

Yaisa smiles like this is not at all the most horrifying thing that has happened since - well, she was going to say 'since Kuthites attacked' but actually getting something wrong which everybody else got right is more horrifying than that, and to top it off she's still incredibly confused about the concept here and imagining scaffolding isn't the helpful kind of advice at all. At...least....she's not going to be punished?

All right, let's try plowing back into how parsing everything into probabilities works.  Once you've practiced these skills hard enough, they become mostly innate and you hardly need to resort to making up numbers in cases where you don't have numbers.  Keltham can think of occasions in the last few days where he's made up numbers, but he's mostly needed to do that because of literally landing in another universe, and then a lot of those occasions would be weird to use as examples.  There's a potential example from a conversation he just had with Asmodia, which is too weird to use, or so Keltham tells the classroom.  And another example just from today's lunch, when Carissa and Ione and Pilar supposedly all went off to get Invisibility copied and it was exactly the three most obviously special girls from a group of eleven girls even though 'copy Invisibility' isn't specialness-laden, but the hypothesis Keltham was actually updating is again too weird to talk about...

Actually, if Keltham thinks back earlier, there's a less weird example.  If you try to find a weird book at Ostenso wizard academy's library, so there's just one local copy, but most people wouldn't usually need to borrow it, what's the chance that it's already checked out when you first try to get it?  Everybody close your eyes, think about the probabilities in your experience, put your hand all the way down for 0, all the way up for 1, closed fist to defy the question.

Why does he keep using examples that are relevant to lies. Maybe because everything's relevant to lies. Is lying here authorized.

Yes, you can imply books are checked out of libraries more often than they in fact are. 

The girls mostly put their hands somewhere around halfway.

The back of Ione's mind is now calculating exactly how unlikely it is to randomly pick those 3 girls from a group of 12 and mostly getting 'really fucking unlikely what the fuck were you thinking' and while that was theoretically Sevar's responsibility to catch and Sevar made up the particular excuse she did, part of Ione still believes deep down that a senior Security is about to have a very unhappy 'conversation' with her about this.

It shows on her face not at all.  She still grew up in Cheliax.

Ione puts her hand a third of the way up; alter-Ione has been checking out weirder books than cleric spell compendiums.

More agreement than he'd have expected?

"Okay to open eyes."

"For the record, my estimate is for the average weird book I check out, not the particular weird book I think you're thinking of."

Heh.  "Right.  Well, this happened when I asked Ione if I could borrow a compendium of cleric spells, and she said that the book existed but was checked out already.  I then considered two possible ways the world could be, and how likely those worlds would be to generate what I observed."

"In one possible world I could have been in, I reasoned, the library actually had plenty of books with all the cleric spells, but those books contained spells that Chelish Governance would rather I not know about, or Ione wasn't sure that was not true given poor general circulation of information around Project Lawful, so she told me there was only one such book and it was already checked out.  If the world is one where Chelish Governance is generally trying to keep me and my capabilities under control, how likely am I to be stonewalled on a book of cleric spells while they try to quickly print up a new one without all the spells they don't want me to know about?  My guess there was around 3/4, or 75% - it's not 100% because maybe they've already got an altered book printed, for example, and in that case they don't have to claim it's not there."

"In the other possible world, there's an ordinary library situation on which Ione reports truthfully.  Then we ask how likely it is that there'd really be only one book with a compendium of half the economically important magic, even if the library is mainly aimed at wizards, and also this book is already checked out.  In dath ilan that'd be very improbable.  Here, I guessed 30%."

Keltham is writing on the white-wall:

P(no-book ◁ Conspiracy) = 75%
P(no-book ◁ Ordinary) = 30%

"Now, what is the Lawful way to think when you find yourself in that situation?  Can you say whether or not I should then think the book was being hidden from me?  Don't bring in all the other facts you know that might be relevant; just as a matter of math, in the Law of Probability, is there anything obvious you can do, any other conclusion you can derive, with the information written on the wall so far?"

Why did she take this job.

(Because she's the only person who even might be able to do it, and that's worth at least three Wishes and a lot of spellsilver, to Hell.)

"It depends on what you considered likelier previously," Meritxell says. "It's likelier than you thought before - though if I were running a conspiracy on you I'd not tell you Ione had magic library powers -"

"Okay Meritxell, that's twice in a row, go to the Overly Advanced Student Holding Cell and use Message next time."

"And yes, that's a fair point about Ione telling me about her magic library powers in the first place, but all that stuff falls into the category of 'please don't bring in all the other facts you know'."

Keltham writes on the white-wall:

Have:

    P(no-book ◁ Conspiracy) = P(no-book & Conspiracy) / P(Conspiracy) = 75%
    P(no-book ◁ Ordinary) = P(no-book & Ordinary) / P(Ordinary) = 30%

"This is as far as we can get by applying the definitions of terms we know, and it doesn't obviously let us derive any further interesting facts or quantities."

"Now, what I want to know is the chance that there's a Conspiracy going on or that things are Ordinary, given that I'm now nearly-certain-except-for-insanity-or-stranger-weirdness, by the way your language needs a shorter word for that, that I was told there was no book of cleric spells available in the Ostenso library.  We could write the quantity I'm interested in as follows:"

Want:

    P(Conspiracy ◁ no-book)
    ----------------------------------
      P(Ordinary ◁ no-book)

"Let's talk a moment about the meaning of that term I just wrote down.  First, in both the numerator and denominator, we're conditioning on being in a world in which I was told there was no book.  Second, starting from inside that world, we narrow it down, in the numerator, to the worlds where there's a conspiracy; and in the denominator, worlds where it's an ordinary library situation and an honest Ione.  By dividing these two numbers, we narrow them down into one number, and that one number is a quantity telling me the relative odds of the Conspiracy World and the Ordinary World; it's how many times more probable the Conspiracy world is than the Ordinary world.  This single quantity might say, for example, 'Conspiracy is twice as likely as Ordinary' or 'Conspiracy is one-third as likely as Ordinary'."

"Why phrase it that way, instead of just asking how likely Conspiracy is in an absolute sense, after observing no-book?  Why not ask for an answer that says, 'If I see no book, a Conspiracy is 20% probable' or some such?  Because to get an absolute probability for Conspiracy given no-book, I'd have to consider every other plausible hypothesis that competes with the Governance conspiracy, including, for example, that Cayden Cailean suddenly mind-controlled Ione to answer falsely, in a way that Governance had nothing to do with."

"By dividing the Conspiracy probability and the Ordinary probability, we can ask about the relative chances there, without dragging in all other possible hypotheses involving say Cayden Cailean."

"Mathematically speaking, what further information do I need to derive that quantity I want, from the quantities I have?"

Need:

    ???

Asmodia quickly expands definitions on her scrap paper, and cancels terms a moment later:

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

Shortly after, Asmodia calls out "Prediction" and Messages Keltham, to which Keltham nods.

Carissa is distracted by being annoyed with Keltham for being too hard to deceive. Why couldn't they have gotten a stupid dath ilani. Why couldn't they have gotten a properly Evil one. 

- if she doesn't learn this stuff and learn it very thoroughly then she's going to lose. She wants to know how many times more likely a conspiracy is if there's no-book. It's not a real problem to do with a real conspiracy, it's just a bunch of meaningless symbols on a page and she needs to cancel some terms and -

"Prediction."

Meritxell is not nearly as sure of how you'd go about this on the math side but she knows the feel of the answer already. the thing you want is how suspicious to be, and obviously to know how suspicious to be you need to know how probable the thing you're suspicious of is, if someone says there's a stray dog that's less suspicious than if they say there's a dragon. And she just has to make the numbers on the paper say the thing that's obviously true. 

"Prediction."