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.