HMMT 十一月 2018 · THM 赛 · 第 5 题
HMMT November 2018 — THM Round — Problem 5
题目详情
- Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is 75% . If Lil Wayne doesn’t make it rain on one day, the probability that he makes it rain the next day is 25% . He decides not to make it rain today. Find the smallest positive integer n such that the probability that Lil Wayne makes it rain n days from today is greater than 49 . 9%.
解析
- Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is 75% . If Lil Wayne doesn’t make it rain on one day, the probability that he makes it rain the next day is 25% . He decides not to make it rain today. Find the smallest positive integer n such that the probability that Lil Wayne makes it rain n days from today is greater than 49 . 9%. Proposed by: Anders Olsen Answer: 9 Let p denote the probability that Lil Wayne makes it rain n days from today. We have p = 0 and n 0 3 1 1 1 p = p + (1 − p ) = + p . n +1 n n n 4 4 4 2 This can be written as ( ) 1 1 1 p − = p − n +1 n 2 2 2 and we can check that the solution of this recurrence is 1 1 p = − n n +1 2 2 1 1 We want < , which first occurs when n = 9. n +1 2 1000