Suppose that I have a biased coin where $\Pr[head] = p$ is very small. Let's say I tossed this coin $n$ times and I saw $h$ heads. Then what is the probability that each pair of heads is separated by at least $\ell$ tails?
For example, let's say $(n, h, \ell) = (10, 2, 5)$.
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There are two heads at location 1 and 9, but they are separated by 6 tails.
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There are two heads at location 6 and 8, but they are separated by only one tail.
One possible approach is summing up all the probabilities of 'yes' events. Is there any closed form for this probability?