# Help with a short paper - cumulative binomial probability estimates

I was hoping someone could help me with a brief statement I can't understand in a book.

The problem I have is with the final line of the following section of Lemma 2.2 (on the second page):

Since $|\mathcal{T}_j|$ is bin. distributed with expectation $n(\log{n})^2 2^{−\sqrt{\log{n}}}$, by the standard estimates, we have that the probability that $\mathcal{T}_j$ has more than $2\mu$ elements is at most $e^{−μ/3} < n^{−2}$. Then, with probability at least $1− \frac{\log{n}}{n^2}$, the sum of the sizes of these sets is at most $n(\log{n})^3 2^{-\sqrt{\log{n}}}$.

Why is this?

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They are only applying it once, to get the $e^{-\mu/3}$. Then they're using a union bound over $\log n$ instances. – Yuval Filmus Feb 19 '12 at 5:29