HMMT 十一月 2019 · 冲刺赛 · 第 17 题

HMMT November 2019 — Guts Round — Problem 17

[10] Kelvin the frog lives in a pond with an infinite number of lily pads, numbered 0, 1, 2, 3, and so forth. Kelvin starts on lily pad 0 and jumps from pad to pad in the following manner: when on lily pad i , he 1 will jump to lily pad ( i + k ) with probability for k > 0. What is the probability that Kelvin lands on k 2 lily pad 2019 at some point in his journey? 3 2

解析

[10] Kelvin the frog lives in a pond with an infinite number of lily pads, numbered 0, 1, 2, 3, and so forth. Kelvin starts on lily pad 0 and jumps from pad to pad in the following manner: when on lily 1 pad i , he will jump to lily pad ( i + k ) with probability for k > 0. What is the probability that k 2 Kelvin lands on lily pad 2019 at some point in his journey? Proposed by: Nikhil Reddy 1 Answer: 2 Suppose we combine all of the lily pads with numbers greater than 2019 into one lily pad labeled ∞ . Also, let Kelvin stop once he reaches one of these lily pads. Now at every leap, Kelvin has an equal chance of landing on 2019 as landing on ∞ . Furthermore, Kelvin is guaranteed to reach 2019 or ∞ within 2020 leaps. Therefore the chance of landing on 2019 1 is the same as missing it, so our answer is just . 2 3 2