HMMT 二月 2017 · 冲刺赛 · 第 23 题

HMMT February 2017 — Guts Round — Problem 23

[ 12 ] Five points are chosen uniformly at random on a segment of length 1. What is the expected distance between the closest pair of points? 2016

解析

[ 12 ] Five points are chosen uniformly at random on a segment of length 1. What is the expected distance between the closest pair of points? Proposed by: Meghal Gupta 1 Answer: 24 Choose five points arbitrarily at a , a , a , a , a in increasing order. Then the intervals ( a − x, a ) , ( a − 1 2 3 4 5 2 2 3 x, a ) , ( a − x, a ) , ( a − x, a ) must all be unoccupied. The probability that this happens is the same 3 4 4 5 5 as doing the process in reverse: first defining these intervals, then choosing five random points none of 5 which lie in the four intervals. This transformed process clearly has a (1 − 4 x ) probability of success. It follows that the desired probability is ∫ 1 / 4 1 5 (1 − 4 x ) dx = 24 0 2016