On the support of a body by a surface with random roughness

Suppose a heavy interval (i.e., an interval in the gravity field) is put on a horizontal line with random roughness. With probability one it is supported at two points, one from the left, and another from the right from its center. We compute probability distribution of support points provided the roughness is fine grained. We also solve an analogous problem where a circle and a disk is put on a rough plane. Some applications in static are given.