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I'm hoping for some help with this excericse in probability.

Let $V$ be a set and let $V'$ be a randomly chosen subset of $V$ such that each element belongs to $V'$ with probability $p$.

Now, let $S \subset V$ such that $|S|= x$, what's the probability that $|V' \cap S|?

It might be easier to assume that $p=1/2$.

Thanks a lot.

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    It will depend on how $V'$ is chosen: if the inclusion/exclusion of each element of $V$ is independent of the others then you may get a different result compared with choosing a fraction $p$ of the elements of $V$.2012-06-08

1 Answers 1

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Community wiki answer so the question can be marked as answered:

As noted by user31373 in a comment, this is the probability of $\lt x/3$ successes in $x$ independent trials with probability of success $p$, which is

$ \sum_{k=0}^{\left\lceil\frac x3\right\rceil-1}\binom xkp^k(1-p)^{x-k}\;. $