HMMT 二月 2024 · 冲刺赛 · 第 13 题

HMMT February 2024 — Guts Round — Problem 13

[9] Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once.

解析

[9] Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once. Proposed by: Albert Wang 149 Answer: 12 Solution: Suppose Mark has already rolled n unique numbers, where 1 ≤ n ≤ 5 . On the next roll, there are 5 possible numbers he could get, with 6 − n of them being new. Therefore, the probability of 6 − n getting another unique number is , so the expected number of rolls before getting another unique 5 5 number is . Since it always takes 1 roll to get the first number, the expected total number of rolls 6 − n 5 5 5 5 5 149 is 1 + + + + + = . 5 4 3 2 1 12