HMMT 二月 2022 · 冲刺赛 · 第 7 题

HMMT February 2022 — Guts Round — Problem 7

[6] A jar contains 8 red balls and 2 blue balls. Every minute, a ball is randomly removed. The probability that there exists a time during this process where there are more blue balls than red balls in the jar can a be expressed as for relatively prime integers a and b . Compute 100 a + b . b

解析

[6] A jar contains 8 red balls and 2 blue balls. Every minute, a ball is randomly removed. The probability that there exists a time during this process where there are more blue balls than red balls a in the jar can be expressed as for relatively prime integers a and b . Compute 100 a + b . b Proposed by: Brian Liu Answer: 209 Solution: One can show that the condition in the problem is satisfied if and only the last ball drown 1 is blue (which happens with probability ), or the blue balls are drawn second-to-last and third-to-last 5 1 1 10 2 (which happens with probability = ). Thus the total probability is = . 10 45 45 9 ( ) 2