# Size of an intersection with a randomly chosen subset

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 $|V' \cap S|<x/3$?

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

Thanks a lot.

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It's the probability of $<x/3$ successes in $x$ independent trials with probability of success $p$. en.wikipedia.org/wiki/Binomial_distribution – user31373 Jun 8 '12 at 18:36
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 $V$. – Henry Jun 8 '12 at 20:05

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