GivenNpoints independently drawn from the uniform distribution on (0, 1), let ᵱnbe the size of the smallest interval that containsnout of theNpoints; let ñpbe the largest number of points to be found in any subinterval of (0, 1) of lengthp.This paper uses a result of Karlin, McGregor, Barton, and Mallows to determine the distribution of ñp, forp= 1/k, kan integer. The paper gives simple determinations for the expectations and variances of ᵱn, for all fixedn> (N+ 1)/2, and of ñ1/2. The distribution and expectation of ñpare estimated and tabulated for the casesp= 0.1(0.1)0.9,N= 2(1)10.