I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$
How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ has the function a Maximum?
Maybe this link might help.
When one computes the ratio $f(x+1)/f(x)$, one sees that many simplifications occur, which make that $f(x+1)\gt f(x)$ is equivalent to $x\lt kp$. Hence $x\mapsto f(x)$ is increasing up to roughly $x=kp$, then decreasing.