I’m afraid your probabilistic reasoning is flawed, Lemur866; in general, you couldn’t hope to justify inductive inference by the rules of probability alone, because the rules of probability are consistent with all kinds of probability distributions.
Specific steps that seem problematic:
Do you mean here to say that I’ve already picked 99 distinct balls out of the jar, with “the last one” being the sole ball I haven’t already picked? The only significant thing that having picked and returned 99 distinct black balls out of a 100-ball jar tells us, in terms of probability, is that the odds of picking a black ball next time are at least 99%… only because the odds are at least 99% that we’ll pick one of those same balls back up again. But it doesn’t automatically tell us anything about the probability of the one so-far unexamined ball being black; indeed, this depends heavily on the (as yet unspecified) probability distribution of barrel contents to begin with [i.e., what was the ratio between the a priori probability of the barrel containing 99 black balls and 1 non-black ball and that of it containing 100 black balls? This can run the full gamut from 0 to 1, and correspondingly will allow the probability we are interested in to run the full gamut from 0 to 1].
Same problem as above.
Same problem as above.
This assumes that all labels are equiprobable, which need not be the case. Again, it depends heavily on the a priori probability distribution of barrel contents our analysis uses. The assumption of equiprobability would also conflict with the style of argument above; after all, there are two possible blackness statuses for the last ball (IS BLACK and ISN’T BLACK), but you felt that (conditional on information about other balls) the probabilities of the two weren’t equal.
Same problem as above (the one about the assumption of all labels being equiprobable). Also, presuming as above that you are speaking here about the probability of a particular label appearing on the sole unexamined ball (rather than the probability of a particular label coming up on the next draw possibly by virtue of the next draw being of a ball already examined), there is no particular upper limit of of 99 (against) to 1 (for) [e.g., it may be the case that the probability distribution of 100-ball jars is such that, with probability 4/5, a jar contain 99 balls labelled “telephone1” through “telephone99”, along with 1 ball labelled “raven”, while, with probability 1/5, a jar contains 100 balls labelled “telephone1” through “telephone100”; in this case, obviously, having drawn 99 distinct non-“raven” balls, the probability of the sole unexamined ball being labelled “raven” is 4/5, much higher than 99 against to 1 for].
This assumes that being non-black and being a raven are not positively correlated. [To see the general problem with analyzing conjunctions by multiplying probabilities without establishing facts about probabilistic correlation, consider: the probability that a die roll gives a non-even number is 1/2 (“1”, “3”, or “5” count). The probability that a die roll gives a prime number is also 1/2 (“2”, “3”, or “5” count). However, the probability that a die roll gives a non-even prime is not less than or equal to 1/2 * 1/2 = 1/4. Rather, it is the higher 1/3 (“3” or “5” count).]
Anyway, to restate my position, I side with you in defending the reasonableness of inductive inference; however, I must take issue with any attempt to justify this from the axioms of probability without further (essentially contrivedly inductive) assumption, as this simply cannot be done: there are plenty of probability distributions which, taken as models of the universe, fail to validate anything like inductive inference.