Random sections of a sphere

The Bertrand paradox is that, whereas we can define in a unique way a point uniformly at random in the interior of a circle, uniformly random chords can be given a variety of competing specifications. This is generalized to spheres, and the distributions of the uniformly random line sections (chords) and plane sections (disks) are tabulated. This includes the large class which are constructed as uniformly random chords of uniformly random disk sections.