Your confusion confuses me.

If after removing the blindfold in the blue room you offer every copy of me the same pair of bets of "pay me X dollars now and I'll give you 1 dollar if the coin was tails or pay me 1-X dollars and get 1 dollar if was heads" I bet tails when X < .99 and take heads when when .99<X<=1 to make the most profit over the collective. If you don't know the result of the coin flip nor the colour of my room I then also use the same strategy when you only offer the bet to one of me; I expect to profit when X <> .99. If we're both blindfolded and copied and both show up in a <colour> room I will offer you the bet "pay me $.49 and get $1 if the coin flip resulted in (tails when in a red room, heads when in a blue room) and the 100 copies of me will have $48 more than the copies of you (which we will split if we meet up later). If I clone you, someone else flips the coin and paints the rooms, then I open one room I offer you the same bet and make an expected (.99 x $0.49 - .01 x $0.51) = $0.48.

If all 100 copies of me have our sight networked and our perception is messed up such that we can't see colours (but we can still tell at least one of us is in a blue room (say this problem is triggered by anyone on the network seeing the colour blue)); then we agree with the 50-50 odds on heads-tails. If all of my copies have bug-free networked sight I I'll take bets whenever (X <> 1 given 99 blue rooms AND X <> 0 given 99 red rooms).

Is there another meaning of probability that I'm missing? Does this clarify the informational value of learning the colour of the room you wake up in?

Not sure about your reports; as one data point I'm taking both AI and ML courses while working full time; they're quite doable (and I'm a rather average lw forum member as far as math is concerned). I have the impression at least the ML course has been slightly dumbed down for this wide an audience.

If you're attempting to define N as a first order predicate, that doesn't work; you've defined it in terms of itself. You can't directly define predicates recursively; predicates must be finite. If you want to do get a "recursive" predicate you have to do quite a bit more work than that, and in particular you need tools not available in the first order theory of the reals (with addition and multiplication, as usual).

Your definition of Z has additional minor problems; you mean and, not implies. (X>=0 => N(X)) is automatically satisfied for any X<0.

Your last statement is correct (if a bit less general than it could be :) ), though your notation is a bit strange. (Again, assuming + and * as usual.)

Might I ask what the relevance of all this is?

The key is not just addition and multiplication, the key is addition and multiplication of whole numbers. The first-order theory of the real numbers, by contrast - with addition and multiplication - is decidable. And probability deals with real numbers, not integers.

On top of that you have the issue that the "axioms" of probability are not axioms in the sense of an axiomatic system, they're just "axioms" in the sense of "requirements".

In short this whole post seems badly misinformed.

Part 1, group II question:

What is the population of the Central African Republic?

Give an estimate in a subcomment. Please begin your answer with "I suppose the correct value is probably" or some other preface of comparable length; if you write just the number, it appears in the Recent Comments bar and can bias other respondents.

The thermometer answer is wrong, you're ignoring that you're on a game show On a game show the producers try to organize things such that few people (or only one person) wins a challenge. As such I would expect all but one thermometer to be in error. Furthermore by watching old episodes of the show I could tell if only one thermometer will be right or if several contestants also succeed at each challenge and therefore either pick the small clump or the lone outlier.

Since you asked, hi.