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Suppose that there are $n$ independent samples $X_1,X_2,...,X_n$ sampled from the uniform distribution on $[0,1]$ with the pdf $f(x)=1$.

Is there a good way to calculate or approximate the probability that $k$ of the $n$ samples are in an interval $[a,a+b]\in[0,1]$? Here only $b$ and $k$ are fixed.

This is like saying there are $k$ samples "clumped" together in an interval of length $b$. Or more mathematically, the event that there exist some interval of length $b$ that covers exactly $k$ samples.

I know that the inclusion-exclusion principle can be used, but I can only get a long and complicated expression using the inclusion-exclusion principle. I'm wondering if there's a better way or if there's a good approximation.

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    Only $b$ is fixed, $a$ can change. I just edited it.2012-04-04

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