I'm looking through a probability derivation, and I see the following inequality.
$$Pr(Y_1 < 0.5n^{0.75} - \sqrt{n}) \leq Pr(|{Y_1 - E[Y_1]}| > \sqrt{n})$$
Is there sufficient information provided here to know where this inequality comes from? $Y_1$ is a binomial random variable. $n^{0.75}$ is the number of trials.
So $E[Y_1] = n^{0.75} p_{success}$.
It is also given that $p_{success} = 0.5 + 0.5/n$
It's easy to show that if $y<0.5n^{0.75}-\sqrt n$, then $|y-(0.5n^{0.75}+0.5n^{-0.25})| > \sqrt n$. This means that the event $Y_1 < 0.5n^{0.75}-\sqrt n$ is contained in the event $|Y_1-E[Y_1]| > \sqrt n$.