For $n>1$ and $~0<p < 1$, can we lower bound the following binomial series in terms of $n$ and $p$ $$\sum_{i=\lceil p n \rceil}^n {n \choose i} (p )^i(1-p)^{(n-i)}.$$
I have posted the question on cs.theroy.stackexchange site, there I came to know that the above can be lower bounded by $$ 1/2- \frac C{\sqrt{p(1-p)n}}.$$ due to Berry–Esseen theorem (for a certain constant C < 0.5). I just wanted to know if a more tighter lower bound is known for the same.
Sorry for repeating the question here.
Thanks in advance!