My question is if there a way to determine the formula or a simplifying form for the following expression: $S = \sum_{i = 0}^{n - 1} {2n \choose i} \cdot x^i \cdot (1 - x)^{2n - i}$
where $0 < x < 1$.
What I noticed so far is that if we denote $S$ as a function of $x$, then we have that:
$S(x) + S(1 - x) = \sum_{i = 0}^{2n} {2n \choose i} \cdot x^i \cdot (1 - x)^{2n - i} = 1$
Unfortunately, I could not find any other relation about $S$.