  In this paper we compare the different phenomena that occur when intersecting
geometric objects with random geodesics on the unit sphere and inside convex
bodies. On the high dimensional sphere we see that with probability bounded
away from zero, the observed length will deviate from the actual measure by at
most a fixed error for any subset, while in convex bodies we can always choose
a subset for which the behavior would be close to a zero-one law, as the
dimension grows. The result for the sphere is based on an analysis of the Radon
transform. Using similar tools we analyze the variance of intersections on the
sphere by higher dimensional random subspaces, and on the discrete torus by
random arithmetic progressions.
