HMMT 十一月 2017 · GEN 赛 · 第 8 题

HMMT November 2017 — GEN Round — Problem 8

Marisa has a collection of 2 − 1 = 255 distinct nonempty subsets of { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } . For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to 8 contain repeated sets.) She repeats this process 2 − 2 = 254 times until there is only one set left in the collection. What is the expected size of this set?

解析

Marisa has a collection of 2 − 1 = 255 distinct nonempty subsets of { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } . For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to 8 contain repeated sets.) She repeats this process 2 − 2 = 254 times until there is only one set left in the collection. What is the expected size of this set? Proposed by: Yuan Yao 1024 Answer: 255 It suffices to compute the probability of each number appearing in the final subset. For any given 7 7 integer n ∈ [1 , 8], there are 2 = 128 subsets with n and 2 − 1 = 127 without. When we focus on only this element, each operation is equivalent to taking two random sets and discarding one of them 128 randomly. Therefore there is a probability that n is in the final subset, and the expected value of 255 128 1024 its size is 8 · = . 255 255 Alternatively, since | A | + | B | = | A ∪ B | + | A ∩ B | , the expected value of the average size of all remaining subsets at a given step is constant, so the answer is simply the average size of all 255 subsets, which is 8 · 128 1024 = . 255 255