Difference between two cumulative distribution functions

Question

Is it possible to rewrite the difference between two binomial cumulative distribution functions, one with parameter $n$ and $p$ and the second with parameter $n$ and $(1-p)$, in such a way that the equation can be written in terms of only one function? Something like that $$ F_{n,p}(x) - F_{n,1-p}(x) = F_{n,p}(x) - (1 - F_{n,p}(x) ) $$

Answer 1

Close. I think it is:

$$F_{n,p}(x) - F_{n,1-p}(x) = F_{n,p}(x) - (1 - F_{n,p}(n-\lfloor x\rfloor - 1)).$$