Many years ago I came across the following task.
If we have the interval $ [0; 1]$ and we throw $ N$ uniformly distributed and mutually independent points on it, then we’ll get $ N+1$ segments. What is the expected length of the longest segment? The 2nd longest? Etc.
For $ N=1$ , the solution is trivial: $ 3/4$ and $ 1/4$ (since the longest segment is uniformly distributed in [1/2; 1] and the shorter one is uniformly distributed in $ [0; 1/2]$ ).
For $ N=2$ , the solution is not trivial, but possible. One just has to draw a quadrat 1 x 1. A point in it would mean that the longest segment has the x coordinate, and the 2nd longest segment has the y coordinate (and the shortest one is 1 – 1st – 2nd). One then has to carefully draw the possible area (this will be a triangle), and find its middle point.
But for $ N>2$ I have no clue how to solve it.
I remember, the book I saw the task in, had a general solution for arbitrary $ N$ , but I don’t know anymore what book it was.
Note that the task is somewhat similar to Average Distance Between Random Points on a Line Segment, but just somewhat.