If we were to randomly drop $n$ needles of random length in a circle, what would be the odds of finding $k$ intersections? This can be asked as:
Randomly place $n$ line segments in a circle. Their length and position is determined by $2$ random points uniformly and independently set in that circle. What are the odds that we will find $k$ intersections?
After seeing the main part of the calculation for what I believe to be the solution for $P(2,1)$, I gave up on calculating $P(n,k)$ because the things behind it are out of my scope (we barely started mentioning integrals).
But I still want to find the $P(n,k)$.
Trying to first calculate the same thing, but with points always being on the circle, seems like a simpler thing to do first: Connecting random points on a circle?