A Threshold for the Size of Random Cover a Sphere

Consider a unit sphere on which are placed N random spherical caps of area 4rip(N). We prove that if lim (p(N).N/log N) < 1, then the probabil i ty that the sphere is completely covered by N caps tends to 0 as N--" 0% and if lim (p(N).N/ log N) > 1, then for any integer n > 0 the probabili ty that each point of the sphere is covered more than n times tends to 1 as N--* oo.