HMMT 十一月 2019 · 冲刺赛 · 第 8 题

HMMT November 2019 — Guts Round — Problem 8

[7] There are 36 students at the Multiples Obfuscation Program, including a singleton, a pair of identical twins, a set of identical triplets, a set of identical quadruplets, and so on, up to a set of identical octuplets. Two students look the same if and only if they are from the same identical multiple. Nithya the teaching assistant encounters a random student in the morning and a random student in the afternoon (both chosen uniformly and independently), and the two look the same. What is the probability that they are actually the same person?

解析

[7] There are 36 students at the Multiples Obfuscation Program, including a singleton, a pair of identical twins, a set of identical triplets, a set of identical quadruplets, and so on, up to a set of identical octuplets. Two students look the same if and only if they are from the same identical multiple. Nithya the teaching assistant encounters a random student in the morning and a random student in the afternoon (both chosen uniformly and independently), and the two look the same. What is the probability that they are actually the same person? Proposed by: Yuan Yao 3 Answer: 17 Let X and Y be the students Nithya encounters during the day. The number of pairs ( X, Y ) for which X and Y look the same is 1 · 1 + 2 · 2 + . . . + 8 · 8 = 204, and these pairs include all the ones in which X and Y are identical. As X and Y are chosen uniformly and independently, all 204 pairs are equally likely to be chosen, thus the problem reduces to choosing one of the 36 pairs in 204, the probability 3 for which is . 17