"We don't know that!  In the - 'object-level' - classroom!"

"So what you're saying here is that if I roll three balls back and forth, and they bounce back and forth a lot, they're equally likely to land in any of the six possible orders, so far as you know?"

"Can that really be true?  After all, there could be bumps in the table, the bouncy walls might not be perfect, I might have rolled the balls harder on some occasions than others..."

"If we don't know any of that on the 'object-level', then, so far as we know, all six permutations are equally likely."

"You could say that sort of thing about any problem with permutations in it."

"Oh, I suppose.  Just don't forget, in real life, always cheat the shit out of anything if you can get any extra information like that."

"After all, if you think otherwise, why, that's the sort of thinking that might lead to you always insisting that all of the chemical elements were equally likely to be in any order on the Periodic Table, any time you don't know for sure what order they have."

"Soooo how about if I then tell you that Ball 2 landed to the RIGHT of Ball 0, again?  What's the probability of RIGHT next time, with the third ball?"

"3/4," says the same person who asked about permutations last time.  (They've unfortunately written their nametag in letters too small to see from where Keltham is standing.)

"This one comes up LEFT, though.  Your prediction FAILED.  All your theories were probably WRONG."

"No, we just ran into a 1/4 chance, that happens sometimes.  Like with the chance of failure on an Augury."

"I suppose.  We should just discard the evidence, then, since it's only to be expected that reality gives us the wrong result, sometimes.  You predict 3/4 RIGHT again next time?"

You can, in fact, get 10-foot range on Prestidigitation, though not when you're a wizard as fresh as Keltham.  One of the students nearest that wall makes their own change to it without leaving their desk.

[----3---0------------1-----2--]

"3/5 right, 2/5 left," they declare.

What if the 2-ball actually landed to the left of the 1-ball?  They're not told which of those was true!

...if those two possibilities are equally probable, then they can sum the probabilities, right?

1/2 chance of 3/5 plus 1/2 chance of 3/5 equals 3/5.

You don't need to label them!

[---|---0---|---|---]

"That's five landing regions, two to the left and three to the right."

Does anyone want to object to the invalidity of that reasoning -

- or alternatively, state a general rule for what to predict on the next round, if you previously got N balls LEFT and M balls RIGHT of the 0-ball, and that's all you know?

"If you have N balls LEFT, there are N+1 landing regions LEFT, and M balls RIGHT gives M+1 landing regions RIGHT, so my first thought is (N+1)/(M+1)."

Ah, yes, so if they have seen one ball go LEFT, and zero balls go RIGHT, the chance of seeing it go LEFT on the next round is (1+1)/(0+1)=2.

So the probability of seeing LEFT next time is 2.  That sounds pretty reasonable to Keltham?  Though clearly anthropics were involved at some point.  Maybe the balls are duplicating people once they start to go LEFT?

They obviously meant, like, a ratio of (N+1):(M+1), not to calculate the probability that way!

So, like... after seeing one ball go right, the chance of seeing it go left on the next round, is half the chance of seeing it go right.  That's what the (N+1)/(M+1) quantity is, how many times more likely LEFT is than RIGHT, in this case, (0+1)/(1+1) = 1/2.

But HOW do they square that up with the previous CLAIM as proven by DUBIOUS INFINITARY MATHEMATICS that, after seeing one RIGHT, the chance of the next ball landing RIGHT is two-thirds?

...it's the same claim.  One-third left is half of two-thirds right.

But is it really.

YES.

Somebody would in fact now like to make the general claim that, if N balls go LEFT and M balls go RIGHT, the chances of the next ball landing LEFT and RIGHT respectively are:

  N + 1           M + 1
---------  vs.  ---------
N + M + 2       N + M + 2

Keltham agrees!

Now would anybody like to try to prove that claim using dubious infinitary mathematics?

"I'd love to, but I don't know where to start... well, actually, I guess the obvious place would be with - two RIGHT and one LEFT and seeing if we can make that come out to 2/5 left and 3/5 right and then seeing if we can get it to generalize."

Several of the new students do in fact know calculus, and that seems like the obvious tool to use on this problem?