Random Intervals
Joyce Justicz, Edward R. Scheinerman, Peter M. Winkler · American Mathematical Monthly · 1990
Fix a large number n and let {x1, y1,…, xn, yn} be 2n points chosen independently from some fixed continuous probability distribution on the real line. Each pair {xi, yi} determines a random interval [min(xi, yi) max(xi, yi)]. We examine the structure of the resulting family of intervals, and in particular answer the following questions: how large a subcollection of pairwise disjoint intervals can one expect to find? And, what is the probability that there is an interval in the family which intersects all the others?