SUMMARY Consider n arbitrary subsets of a set X. Suppose x e X lies in H(x) of these subsets and define H = min H(x) and H = max H(x). That is, the least and most covered regions of X are x E X _ x E respectively H- and H-covered, with 0 n-rm) are determined for m = 0, 1,. 1. INTRODIUCTION The probability p(n, 0) that the surface 99 of a sphere is completely covered by n random circular caps of equal angular radius 6 (O 0 N(6), 1 -p(n, 0) = 4n(n - 1) {U(n)}-1 sin2 (10) COS2n-2 (10), (1) where U(n) is the expected number of uncovered crossings between perimeter circles, given