Finding value of $p$ which satisfies Binomial probability? Time/space considerations

Question

I wish to find the value of $p$ between 0 and 1 which satisfies:

$$P[\operatorname{Bin}(n-1, p) < k] = \frac k n$$

Here $k$ and $n$ are integers with $k < n$.

Also, how many computations in terms of $n$ and $k$ would it take for me to reach the solution?