Let $X\sim Bin(n,p)$ and $c>0$.
Confidence intervals for $X$, such as Chernoff's, have the form $$\Pr\left[|X-np|\ge a\right]\le b.$$
I'm looking for a (nice to work with) lower bound.
Namely, how can we find $\delta_{min}$ such that $$\Pr\left[|X-np|\ge c\right]\ge \delta_{min}?$$
Obviously, we have $\delta_{min}=1-\sum_{i=np-c}^{np+c}{n\choose i}p^i(1-p)^{n-i}$.
Is there a nice closed-form expression that bounds this from below?