Problem 398

Problem 398

Cutting rope

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(107, 100).
Give your answer rounded to 5 decimal places behind the decimal point.



记E(n, m)为第二短的分段的期望长度。
例如,E(3, 2) = 2以及E(8, 3) = 16/7。

求E(107, 100)。