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?
Uniform sampling of multicolored balls
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probability
sampling
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2For 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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0Let'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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0Yes; you just can’t guarantee it. – 2011-10-28
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1Yes, for appropriate definitions of "expect" and "roughly the same". – 2011-10-29
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0Which would be...? – 2011-10-29
1 Answers
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Taking your example "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" then for any colour which has a fraction $k$ of the balls in the big urn will with probability greater than 99.5% have a fraction in the small urn in the range $[k-0.001,k+0.001]$.
If $k$ is very small or very large then the range can be narrowed or the probability increased, but any answer will be of this type.