They are not immediately sure. 

"I mean, you can derive a bunch of silly things about the probability of all other events?"

"Yeah, like, things get more probable if you assert that a thing and the probability three event happen - wait, do they - that's not on the board but I'm pretty sure it's true -"

Why, nonsense, the probability 3 event and the regular event could be the sort of things that never happen at the same time, in which case the probability of their intersection is 0.  Say, the chance that Keltham is holding a silver coin is 0.5, and the chance that Keltham is not holding a silver coin is 3.  The chance that Keltham is holding and not holding a silver coin is surely 0.

No, that breaks the rule on the board about how the probability of a thing and not-that add to one.

Pffft, rules.  What good do rules ever do anyone?  Rules just stop people from doing what they want, and are therefore, universally, bad.  If we're going to violate the rule about probabilities being between 0 and 1, let's violate the sum-to-1 rule too!  Who needs that rule - what bad thing happens to you if you violate that one?

You can't just go about justifying rules by appealing to other rules, there has to be a reason why anyone cares about any of the rules in the first place!

The inevitable collapse of Asmodean sanity seems kind of hopeless for Ione to stop, actually?  Once you're paying attention and your ears have been attuned to listen, you start to notice how Project Lawful may be the most intrinsically doomed thing that has ever been tried in the history of Golarion.

The reason for the rules is that they're the only set of rules that have the useful properties discussed earlier in the lecture?

Nice try at guessing the password on his Forbiddance, but that's not how dath ilani education works.  Which of those useful properties fails at probability 3, and how?

Asmodia has been thinking about it.  "Someone's going to try saying that it means you can gain 2s instead of losing 2s in the scoring game, you're going to say so what -"

"Gonna say it was a very depressing game and the prospect of being able to gain anything at all in it sure sounds nice, and, yes, I was waiting for somebody to try that line on me."

"After which Pilar comments that she liked the game the old way -"

...okay fine yes she was thinking that, so she's a good Asmodean, sue her.

"But mostly I think it will appeal to some principle you haven't shown us yet.  That'd be my prediction."

"Well, invent the principle then."

"There's at least two lessons here.  The first lesson is noticing when you have no explicit idea why a set of rules has to be the way it is, and couldn't give a strong solid answer about what goes wrong, if somebody asks you, well, how about if the rules were different, how about if we break those rules?  It's closely related to the art of making sure that your beliefs mean anything.  The way of having math mean anything, if you are using that math for something and not just admiring it, is to say what ill fate would befall you if you used different math."

"And the second lesson is, Civilization didn't get to learn the principle that, yes, I haven't covered yet, by successfully noticing they didn't know it, and then sitting back in class waiting for a teacher or a god to tell them.  The ancestors of Civilization noticed the gap in their knowledge, some people tried to fill it in, somebody eventually succeeded, and that's why Civilization now knows.  You try.  Come up with some bad thing that happens to people who assign a 300/100 chance that something happens.  I'm not going to answer until I see somebody try, even if they fail, because I'd rather teach people to try and fail, then to teach them to wait for the teacher to answer."

"Say there's a merchant ship, and you make 100 gold if it comes back and 0 gold if it doesn't, how much should you be willing to spend to send it? If it has a .5 chance, 50 gold - eliding that you want a profit, for the moment - If it has a 0 chance, 0 gold. If it has a 3 chance....300 gold?"

"Go to the Overly Advanced Student Holding Cell next to Asmodia and use Message next time you're that far ahead.  Nobody else learns things if they just wait for Carissa to tell them."

"She has a headband," someone mutters.

She does and she loves it. 

"And now she's in the Overly Advanced Student Holding Cell, so you can continue to learn things in advance of your own headbands arriving.  Also we should now have Fox's Cunnings to use if we run into a stumbling block, and I'm mostly thinking we should save them for more technical stumbling blocks, but if you feel like you have a thought and can almost complete it, you're allowed to request one, or use one of your own spells if you've got a Cunning hung."

"But to generalize the point beyond merchant ships, the purpose for which we ultimately use our wordless senses of which things are more or less likely, is to divide up our few resources between the many things we could potentially try to do.  Not just limited resources like money, but limited time, limited attention; you can only act so often and only think so much."

"Tomorrow will go differently depending on whether Nidal has figured out our new location and attacks us again even with their god sealed up, or if, alternatively, we have an enjoyable day of lectures and me learning some magic and other activities.  In the latter case, Fox's Cunnings are quite useful.  In the former case, where Nidal attacks, Fox's Cunnings are less useful and spells for setting things on fire are more useful.  But again if it's a peaceful day, spells for setting things on fire are less useful."

"So how can we possibly plan, in such an uncertain world?  The best strategy if the universe is one way, is not at all the best strategy if the universe is a different way!  And we are uncertain and can never attain absolute certainty because it's not the sort of thing that's true in every possible world, and if it were we might make a logic error anyways.  Oh woe!  Oh alas!  Shall we just choose at random since no choice is perfectly defensible?"

"And the answer is, you put weights upon the different worlds that might be true, and figure out the consequences in those different worlds, and weigh those consequences according to their probability.  A thing that makes the scale from 0 to 1, more useful than the earlier scale from 1-12, is that 0 reflects not being at all concerned about the consequences to us in some world, because we think that reality just can't be that way, we multiply the weight of consequences there by 0 hundredths out of 100.  1 says that we're going to weigh the consequences only if that proposition is true, because we are certain of it; we're going to take all of those consequences as objects of concern, and not diminish them at all in proportion to their uncertainty and unlikelihood.  To be clear, we are never actually certain; even Nethys, Ione told us, reasons in probabilities, just more extreme ones.  But sometimes we are sure enough that it's not worth the cost of thought to weigh the other possibilities more finely."

"A probability of 3 is like saying that the consequences of something weigh on your mind three times as much, getting three times the share of your limited resources, as some possibility you were absolutely certain of; and if you try to cash out what that could even mean, you start getting results like the one Carissa talked about - that you spend 300 gold with certainty, in order to make back 100 gold with probability 3, because that outcome weighs three times as much in your calculations as if it were certain."

"Among the ways that we'd make our way to the corresponding law-Fragment if we were doing everything in slow careful order, would be to show that, if we're not going to end up going in circles in certain ways, we need to consistently weigh the possibilities we deal with, and make choices based on weights of consequences, and these weights of consequences end up looking like multiplying the consequences by their probability.  And then having gone through all that slow pathway, you'd be able to say, more generally than in the special case of merchant ships that have costs and profits measurable in gold, what ill fates will befall someone if they weigh a consequence three times more strongly than if it were certain."

"That's the principle you're missing, in this case.  I'm not actually going to cover that Law-fragment at least today, but I wanted to at least make clear what I was skipping; because someday in due time you're going to have to redesign Chelish education, and it's going to have to go back and fill in, not just that part, but all the other Law-fragments we're skipping that pin down and spotlight the Law of Probability."

"Or to summarize, and maybe to see it at a glance:  To see what bad thing would happen to you, if you assigned a probability of 3, you'd have to be using that probability for something and doing something with that probability, such as, for example, sending out merchant ships, or making a bet, or deciding how many of your limited spell slots to spend on preparing against a Nidal attack."

Nods all around. 

"Also if 300 out of 100 people with INT 15 could become fifth-circle wizards, you'd start with a graduating class of 150 INT 15s and get 450 fifth-circle wizards out of them.  Can I point to that example and say it's not allowed - is that an instance of the same principle?"

"Interesting question!  And you might ask then, what if there's magic that can make three copies of somebody?  Maybe you just can start with 150 INT 15s and get 450 5th-circle wizards.  And, if so, what would it be like to be one of those wizards?  Should you then go around saying that your chance of making 5th-circle is 300%, and weigh the consequences of that three times as much as if you were certain?"

"This however would get us into 'anthropics', and we are not getting into 'anthropics'.  That, by the way, is a general slogan of dath ilani classes on probability theory:  We are not getting into 'anthropics'.  I'm not even going to translate the word.  As long as you don't make any copies of people, you can stay out of that kind of trouble.  That trouble-free life should be our ambition for at least the next several weeks."

"I'm sorry I asked."

"Under ordinary circumstances I'd say not to be sorry for asking, that's how people learn, but you did ask about 'anthropics' so instead I'll say, apology accepted."

Shit now she urgently needs to figure out what an anthropics is. Any abstract concept that Keltham is jokingly terrified of is going to be relevant to her life very soon, it's getting to be a rule.