ON THE PROBABILITY OF COVERING THE CIRCLE BY RANDOM ARCS

Arcs of length lk, 0 < lk < 1, k = 1, 2, * , n, are thrown independently and uniformly on a circumference W having unit length. Let P(11, 12, * * *,) be the probability that W is completely covered by the n random arcs. We show that P(I?, 12,*. , 14) is a Schur-convex function and that it is convex in each argument when the others are held fixed. COVERAGE PROBABILITIES; SCHUR-CONVEX; GEOMETRICAL PROBABILITY