HMMT 二月 2024 · 冲刺赛 · 第 17 题

HMMT February 2024 — Guts Round — Problem 17

[11] The numbers 1 , 2 , . . . , 20 are put into a hat. Claire draws two numbers from the hat uniformly at random, a < b , and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, c < d . Let N denote the number of integers n that satisfy exactly one of a ≤ n ≤ b and c ≤ n ≤ d . Compute the probability N is even.

解析

[11] The numbers 1 , 2 , . . . , 20 are put into a hat. Claire draws two numbers from the hat uniformly at random, a < b , and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, c < d . Let N denote the number of integers n that satisfy exactly one of a ≤ n ≤ b and c ≤ n ≤ d . Compute the probability N is even. Proposed by: Rishabh Das 181 Answer: 361 Solution: The number of integers that satisfy exactly one of the two inequalities is equal to the number of integers that satisfy the first one, plus the number of integers that satisfy the second one, minus twice the number of integers that satisfy both. Parity-wise, this is just the number of integers that satisfy the first one, plus the number of integers that satisfy the second one. 10 The number of integers that satisfy the first one is b − a + 1 . The probability this is even is , and 19 9 odd is . This means the answer is 19 2 2 10 + 9 181 = . 2 19 361