HMMT 十一月 2020 · 冲刺赛 · 第 28 题

HMMT November 2020 — Guts Round — Problem 28

[15] Bernie has 2020 marbles and 2020 bags labeled B , . . . , B in which he randomly distributes the 1 2020 marbles (each marble is placed in a random bag independently). If E the expected number of integers 1 ≤ i ≤ 2020 such that B has at least i marbles, compute the closest integer to 1000 E . i

解析

[15] Bernie has 2020 marbles and 2020 bags labeled B , . . . , B in which he randomly distributes 1 2020 the marbles (each marble is placed in a random bag independently). If E the expected number of integers 1 ≤ i ≤ 2020 such that B has at least i marbles, compute the closest integer to 1000 E . i Proposed by: Benjamin Kang Answer: 1000 Solution: Let p be the probability that a bag has i marbles. Then, by linearity of expectation, we i find E = ( p + p + · · · ) + ( p + p + · · · ) + · · · = p + 2 p + 3 p + · · · . 1 2 2 3 1 2 3 This is precisely the expected value of the number of marbles in a bag. By symmetry, this is 1.