HMMT 十一月 2018 · 冲刺赛 · 第 27 题

HMMT November 2018 — Guts Round — Problem 27

[ 13 ] At lunch, Abby, Bart, Carl, Dana, and Evan share a pizza divided radially into 16 slices. Each one takes takes one slice of pizza uniformly at random, leaving 11 slices. The remaining slices of pizza form “sectors” broken up by the taken slices, e.g. if they take five consecutive slices then there is one sector, but if none of them take adjacent slices then there will be five sectors. What is the expected number of sectors formed? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2018, November 10, 2018 — GUTS ROUND Organization Team Team ID# th th

解析

[ 13 ] At lunch, Abby, Bart, Carl, Dana, and Evan share a pizza divided radially into 16 slices. Each one takes takes one slice of pizza uniformly at random, leaving 11 slices. The remaining slices of pizza form “sectors” broken up by the taken slices, e.g. if they take five consecutive slices then there is one sector, but if none of them take adjacent slices then there will be five sectors. What is the expected number of sectors formed? Proposed by: Andrew Gu 11 Answer: 3 Consider the more general case where there are N slices and M > 0 slices are taken. Let S denote the number of adjacent pairs of slices of pizza which still remain. There are N M slices and a sector of k slices contributes k 1 pairs to S . Hence the number of sectors is N M S . We compute the expected value of S by looking at each adjacent pair in the original pizza: N 2 ( N M )( N M 1) ( N M )( N M 1) M E ( S ) = N = N = N N ( N 1) N 1 M The expected number of sectors is then ( N M )( N M 1) ( N M ) M N M = . N 1 N 1 11 For N = 16 , M = 5 this yields . 3 th th