Take the Unit Circle. Let U1 and U2 be two randomly distributed points on the Unit Circle, and let L1 be the line that connects them. Let V1 and V2 be two additional randomly distributed points on the unit circle, and let L2 be the line that connects them.
What is the probability that L1 intersects L2?
I've got an idea of how this might work, but I'm not sure. You can map the Unit circle to a straight line of length 2pi, and maybe arbitrarily rotate so that U1 is at 0. Then U2 is a uniform random variable from 0 to 2pi. Is this good so far?
Then, L2 will only intersect L1 if either V1 < U2 < V2 or V2 < U2 < V1. If V1, V2 > U2 or V1, V2 < U2 then the lines will not intersect when mapped to a circle. Therefore, we need to find the P( V1, V2 > U2) + P(V1, V2 < U2). Again, does this seem correct?
That's where I'm not sure how to continue. Can anyone comment on my approach to the problem so far and where I should go from here? There's also a more general question but I can save that for later.