Given number $0<p<1$, I want to compute for any integer $n>0$ the probability that at most $p\cdot n$ coins are head. I know that I can sum
$${n \choose 0} + {n \choose 1} + \ldots + {n \choose \lfloor p\cdot n \rfloor}$$
and divide it by $2^n$. But I am interested in a more "direct" estimate, from which I can see (for fixed $p$), how this number evolves with growing $n$.