HMMT 二月 2012 · 冲刺赛 · 第 5 题

HMMT February 2012 — Guts Round — Problem 5

[ 3 ] Mr. Canada chooses a positive real a uniformly at random from (0 , 1], chooses a positive real b uniformly at random from (0 , 1], and then sets c = a/ ( a + b ). What is the probability that c lies between 1 / 4 and 3 / 4?

解析

[ 3 ] Mr. Canada chooses a positive real a uniformly at random from (0 , 1], chooses a positive real b uniformly at random from (0 , 1], and then sets c = a/ ( a + b ). What is the probability that c lies between 1 / 4 and 3 / 4? Answer: 2/3 From c ≥ 1 / 4 we get a 1 ≥ ⇐⇒ b ≤ 3 a a + b 4 and similarly c ≤ 3 / 4 gives a 3 ≤ ⇐⇒ a ≤ 3 b. a + b 4 Choosing a and b randomly from [0 , 1] is equivalent to choosing a single point uniformly and randomly from the unit square, with a on the horizontal axis and b on the vertical axis: b = 3 a a = 3 b To find the probability that b ≤ 3 a and a ≤ 3 b , we need to find the area of the shaded region of the square. The area of each of the triangles on the side is (1 / 2)(1)(1 / 3) = 1 / 6, and so the area of the shaded region is 1 − 2(1 / 6) = 2 / 3. Guts