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If I uniformly sample-without-replacement a small bunch of multicolored balls (say, five colors) from one urn into a "smaller" urn, will the distribution of ball colors in the smaller urn be the same as the true distribution of ball colors in the larger urn, within some interval?

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    For certain? Only if the interval is large enough to include every possible sample, no matter how unrepresentative. The sample size $n$ is also relevant: you can’t get a very representative sample if $n$ is just $1$ or $2$, say.2011-10-28
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    Let's say I have a population of 10M balls. I close my eyes and pull out 5% of them from this very large urn, which is 500K balls. Can I expect that the 5% subset I pulled out has roughly the same distribution of colors as the larger population?2011-10-28
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    Yes; you just can’t guarantee it.2011-10-28
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    Yes, for appropriate definitions of "expect" and "roughly the same".2011-10-29
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    Which would be...?2011-10-29

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