4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim round sphere. Here uniformly is in the obvious sense of the volume form associated with the round metric. What is the probability the two curves cross?
The probability is 1/8. With each (generic) pair of geodesic segments you can associate another 15 pairs which you get by replacing each of the 4 end points by its antipodal point. You can easily check that of these 16 pairs, exactly 2 intersect.
If you extend to complete circular geodesics then the probability is 1. So it is just a matter of finding the probability that the point of intersection is in both segments chosen, which is an easier linear (not spherical) problem.