OK, and I can illustrate this with the ball example again.
Suppose I have a jar with 100 balls, and I allow you to pull out balls at random and observe their color and label. You draw out 99 balls and observe that none of them are black, and they are labled various things…“apple”, “telephone”, “car” and so on, but none of them are labeled “raven”.
I then ask you to wager about the remaining ball. What are the odds that the remaining ball is not black but is labeled “raven”?
We know that at least 99 balls out of 100 were not black. If there were one black ball, the odds that it would be the last one picked are 99 to 1. Maybe there weren’t any black balls in the jar, but the odds that the last one is not black is at least 1 to 99.
Similarly, we know that at least 99 balls out of 100 were not labeled “raven”. So the odds that the last ball is labeled raven is AT LEAST 99 to 1. Since the labels have been all kinds of things, we have no idea what the last ball will turn out to be labeled…but we’re pretty confident it isn’t “raven”. And if we haven’t seen “soup” or “jam” we’re also exactly as confident that the last ball isn’t soup or jam. If we hypothesize that there are an infinite number of potential labels, then any label has a zero possibility. But for any real world label we’ve only got a finite number of possibilities. So infinity is out, but we’ve got some large but unknown number of possibilities. But since there were only 100 actual balls, the maximum possible odds for that one remaining ball is 99 to 1 for any potential label. It could be much smaller than that, but it’s at least 99 to 1.
So what are the maximum odds that the last ball is a non-black raven? .99*.01, or .0099. In other words, if I offered you odds better than 1 to 99, you should take the bet that the last ball won’t disprove the hypothesis that all ravens are black.
The reason looking at 99 black ravens and guessing that the 100th raven will also be black seems reasonable, while looking at 99 nonblack nonravens and guess that the 100th thing will also be a nonblack nonraven seems unreasonable, is that there are a lot more than 100 things in the world.
If there were only 100 things in the world, finding 99 nonblack nonravens would be just as good of evidence that all ravens are black as finding 99 black ravens. We have an intuitive sense that a sample of 100 ravens is a nontrivial sample of ravens–but a sample of 100 nonblack things is a pretty poor sample of nonblack things.