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Originally Posted by snarkophilus
Carter claims that a 10 ball urn is more likely than a million ball urn. That may be true, but it is not more likely than an urn with between a million and fifty million balls. And so it goes. You can always find a range of urns that is more likely than the last. And that's where his error lies.
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Two problems:
1) If your purpose is to find a 7, then of course sampling from a large number of large populations will give you more chance than sampling from a single small population. But that's irrelevant to our problem: it's one ball, and one bite at the cherry, whereas your summing of independent probabilities implies that you're sitting with a big range of urns, pulling a ball out of each, and counting success if
any of those balls is a 7.
2) Carter's not interested in the unlikelihood of pulling a
particular number, but in the unlikelihood of our selected number (
whatever it is) coming from the lowest 5% of the numbered population.
I've already rehearsed Carter's calculation earlier in the thread:
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A ball chosen at random has only a 5% chance of coming from the lowest-numbered 5% of balls, and a 95% chance of coming from the other, higher-numbered balls. We are therefore 95% certain that our ball, number 7, has a higher number than the lowest 5% of balls. So we are 95% certain that the lowest-numbered 5% consists of fewer than 7 balls. If there are only 6 or fewer balls in a 5% sample, then the total number of balls must be (6*20)=120 or fewer. When we draw ball number 7, we are therefore immediately 95% confident that there are 120 or fewer balls in the urn.
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Grant Hutchison
Edit: Slight expansion for clarity.