# Uniform sampling of multicolored balls

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. – Brian M. Scott Oct 28 '11 at 23:29
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? – Alex Reynolds Oct 28 '11 at 23:33
Yes; you just can’t guarantee it. – Brian M. Scott Oct 28 '11 at 23:56
Yes, for appropriate definitions of "expect" and "roughly the same". – mhum Oct 29 '11 at 0:31
Which would be...? – Alex Reynolds Oct 29 '11 at 1:49

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.