Upper and lower bound of binomial summation For $n&gt;1$ and $~0&lt;p \leq 1$,   can we upper and lower bound the following   binomial series in terms of $n$ and $p$ $$\Sigma_{i=\lceil p n \rceil}^n {n \choose i} (p )^i(1-p)^{(n-i)}$$
Let me denote the sum as $S_{n,p}$. It gives the probability that a random sample from the binomial distribution $B(n,p)$ exceeds its expected value.

For $0&lt;p&lt;1$ constant, the central limit theorem tells you that $B(n,p)$ looks like the normal distribution $N(pn,p(1-p)n)$ for large $n$. Since this is symmetric, $S_{n,p}$ will be approximately $1/2$ as noted in usul’s comment.

This can be quantified using the Berry–Esseen theorem: applying it to $n$ indicator random variables with mean $p$, variance $p(1-p)$, and third moment $(p^2+(1-p)^2)p(1-p)\le p(1-p)$, gives

$$\left|S_{n,p}-\tfrac12\right|\le\frac C{\sqrt{p(1-p)n}}$$

for a certain constant $C&lt;1/2$.