Tight bound of random interval packing

Consider n random intervals I-1, . . . , I-N chosen by selecting endpoints independently from the uniform distribution. A packing of I-1, . . . , I-N is a disjoint sub-collection of these intervals: its wasted space is the measure of the set of points not covered by the packing. We investigate the random variable W-N equal to the smallest wasted space among all packings. Coffman, Poonen and Winkler proved that EWN is of order (log N)(2)/N. We provide a sharp estimate of log P(W-N greater than or equal to t(log N)(2)/N) and log P (W-N less than or equal to t(log N)(2)/N) for all values of t.