HMMT 二月 2009 · 冲刺赛 · 第 11 题

HMMT February 2009 — Guts Round — Problem 11

[ 7 ] There are 2008 distinct points on a circle. If you connect two of these points to form a line and then connect another two points (distinct from the first two) to form another line, what is the probability that the two lines intersect inside the circle?

解析

[ 7 ] There are 2008 distinct points on a circle. If you connect two of these points to form a line and then connect another two points (distinct from the first two) to form another line, what is the probability that the two lines intersect inside the circle? Answer: 1 / 3 Solution: Given four of these points, there are 3 ways in which to connect two of them and then connect the other two, and of these possibilities exactly one will intersect inside the circle. Thus 1 / 3 of all the ways to connect two lines and then connect two others have an intersection point inside the circle. 2