HMMT 二月 2024 · 冲刺赛 · 第 2 题

HMMT February 2024 — Guts Round — Problem 2

[5] Jerry and Neil have a 3 -sided die that rolls the numbers 1 , 2 , and 3 , each with probability . Jerry 3 rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry’s. Compute the probability that Neil’s final number is 3 . n

解析

[5] Jerry and Neil have a 3 -sided die that rolls the numbers 1 , 2 , and 3 , each with probability . Jerry 3 rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry’s. Compute the probability that Neil’s final number is 3 . Proposed by: Rishabh Das 11 Answer: 18 1 Solution: If Jerry rolls k , then there is a probability that Neil’s number is 3 , since Neil has an 4 − k equal chance of rolling any of the 4 − k integers not less than k . Thus, the answer is ( ) 1 1 1 11 1 + + = . 3 2 3 18 n