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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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.