Eccoci.

Inside a rope of length $n$, $n - 1$ points are placed with distance $1$ from each other and from the endpoints. Among these points, we choose $m - 1$ points at random and cut the rope at these points to create $m$ segments.

Let $E(n, m)$ be the expected length of the second-shortest segment. For example, $E(3, 2) = 2$ and $E(8, 3) = 16/7$. Note that if multiple segments have the same shortest length the length of the second-shortest segment is defined as the same as the shortest length.

Find $E(10^7, 100)$. Give your answer rounded to $5$ decimal places behind the decimal point.