1. In the first of those cases, with the biased coin, suppose that in the second phase you flip it _thousands_ of times and it comes up with N+1 heads and N tails. Then the statistical significance of the overall bias can become arbitrarily small, and if you had a concrete alternate hypothesis along the lines of "at least 5% more likely to come up H than T" the overall evidence can be extremely strong evidence against that.
2. In the second case, with the coloured balls, note that we can readily make each bit of evidence for A > B > C much stronger than the 2 : 1 : 0 above; the bag in hypothesis B could be, say, 1% red, 1% blue, 98% green. We'd still end up with B being the only one of our viable possibilities after both observations.
3. There are real-world situations with the sort of structure we see in that second case. Suppose hypothesis A is "there is a god who not infrequently works miracles", hypothesis B is "there are no gods, but people not infrequently fabricate evidence of miracles", and hypothesis C is "nothing at all miraculous-looking ever happens". Then you see what looks like a convincing case of evidence for one god, and then what looks like a convincing case of evidence for a different one. The two gods are from different religions each of which adamantly insists that the other one is pure invention.
4. The real world is never as neat as these examples. E.g., in the gods scenario you also need to consider possibilities like "actually both those religions have some truth to them, and rivalry between their gods is why each one's followers insist that the other religion is completely imaginary" and "there is a miracle-working god, but also followers of other gods fabricate miracles" and "there are no gods and very little fabrication, but sometimes incredibly unlikely things happen just by chance" and "there are no gods but there are super-advanced aliens who enjoy messing with us" and so forth. Even in the balls-from-bags scenario -- if it happens in the real world rather than in a textbook -- you have to consider things like "I was mistaken about what ball I picked the first time" and "actually sometimes one blue ball gets into a big bag of red balls by accident" and so forth.
(A lot of the most important work in, e.g., science consists not of testing rival hypotheses but of figuring out viable hypotheses to test.)
Comments
A couple of other remarks.
1. In the first of those cases, with the biased coin, suppose that in the second phase you flip it _thousands_ of times and it comes up with N+1 heads and N tails. Then the statistical significance of the overall bias can become arbitrarily small, and if you had a concrete alternate hypothesis along the lines of "at least 5% more likely to come up H than T" the overall evidence can be extremely strong evidence against that.
2. In the second case, with the coloured balls, note that we can readily make each bit of evidence for A > B > C much stronger than the 2 : 1 : 0 above; the bag in hypothesis B could be, say, 1% red, 1% blue, 98% green. We'd still end up with B being the only one of our viable possibilities after both observations.
3. There are real-world situations with the sort of structure we see in that second case. Suppose hypothesis A is "there is a god who not infrequently works miracles", hypothesis B is "there are no gods, but people not infrequently fabricate evidence of miracles", and hypothesis C is "nothing at all miraculous-looking ever happens". Then you see what looks like a convincing case of evidence for one god, and then what looks like a convincing case of evidence for a different one. The two gods are from different religions each of which adamantly insists that the other one is pure invention.
4. The real world is never as neat as these examples. E.g., in the gods scenario you also need to consider possibilities like "actually both those religions have some truth to them, and rivalry between their gods is why each one's followers insist that the other religion is completely imaginary" and "there is a miracle-working god, but also followers of other gods fabricate miracles" and "there are no gods and very little fabrication, but sometimes incredibly unlikely things happen just by chance" and "there are no gods but there are super-advanced aliens who enjoy messing with us" and so forth. Even in the balls-from-bags scenario -- if it happens in the real world rather than in a textbook -- you have to consider things like "I was mistaken about what ball I picked the first time" and "actually sometimes one blue ball gets into a big bag of red balls by accident" and so forth.
(A lot of the most important work in, e.g., science consists not of testing rival hypotheses but of figuring out viable hypotheses to test.)