HMMT 十一月 2021 · 冲刺赛 · 第 2 题

HMMT November 2021 — Guts Round — Problem 2

[5] There are 8 lily pads in a pond numbered 1 , 2 , . . . , 8. A frog starts on lily pad 1. During the i -th 1 second, the frog jumps from lily pad i to i + 1, falling into the water with probability . The probability i +1 that the frog lands safely on lily pad 8 without having fallen into the water at any point can be written m as , where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . n

解析

[5] There are 8 lily pads in a pond numbered 1 , 2 , . . . , 8. A frog starts on lily pad 1. During the 1 i -th second, the frog jumps from lily pad i to i + 1, falling into the water with probability . The i +1 probability that the frog lands safely on lily pad 8 without having fallen into the water at any point m can be written as , where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . n Proposed by: Joseph Heerens Answer: 108 Solution: The probability the frog lands safely on lily pad i + 1 given that the frog safely landed on i lily pad i is . The probability the frog make it to lily pad 8 safely is simply the product of the i +1 probabilities of the frog making it to each of the lily pads 2 through 8 given it had safely landed on the lily pad before it. Thus, the probability is 1 2 7 1 · · · · = . 2 3 8 8