Assume that we know
$$\exp(-\frac{t^2/2}{np(1-p)+t/3})$$
and our goal is to obtain which value of $t$ we can take such that the above is bounded by $n^{-\gamma}$, for certain constant $\gamma $.
The answer is to take $t=\sqrt{2\gamma np(1-p)\log n}+2\gamma \log n/3$.
But I have idea to obtain it. Could someone help me out? Thanks!