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.

shortestgeodesics, in which case the answer is not so immediate, though not very hard either. $\endgroup$ – Wlodek Kuperberg Jan 6 '14 at 15:41