PUMaC 2017 · 团队赛 · 第 4 题
PUMaC 2017 — Team Round — Problem 4
题目详情
- (4) Ayase chooses three numbers a, b, c independently and uniformly from the interval [ − 1 , 1]. p The probability that 0 < a + b < a < a + b + c can be expressed in the form , where p and q q are relatively prime positive integers. What is p + q ?
解析
- 0 < a ⇒ a > 0, a + b < a ⇒ b < 0, and a + b < a + b + c ⇒ c > 0. Each of these conditions 1 happens with probability . Also, note that given a > 0 and b < 0, 0 < a + b ⇒ | a | > | b | . Also, 2 a + b + c > a ⇒ b + c > 0 ⇒ | c | > | b | . The probability that | b | is less than both | a | and | c | is 1 1 1 1 1 1 ; consider ordering the 3 in a row. Hence, the final answer is · · · = ⇒ p + q = 25 . 3 2 2 2 3 24 Problem written by Bill Huang