折断木棍成三角形
Stick to Triangle
题目详情
在一根长度为 1 的木棍上随机选择两个断点，把木棍折成三段。问这三段能组成三角形的概率是多少？
A stick is broken into 3 parts, by choosing 2 points randomly along its length. With what probability can it form a triangle?
Hint
All three broken parts must satisfy the triangle inequality. Or rather, each of the broken part must be less than half of stick's length.
解析
答案是 。
能成三角形当且仅当三段长度都小于 （否则最长边不小于其余两边之和）。在单位正方形参数空间中满足条件的区域面积占比为 。
Original Explanation
1/4
Solution
All 3 sides have to have lengths less than half the length of the stick. the conditions are min{ x.y}<= 0.5; max{x,y}>=0.5; |x-y|<=0.5 . looking at the unit square, and dividing into 8 congruent triangles by lines parallel to the axes and y=x line, its easy to see 2 of the 8 triangles satisfy the condition. so the answer is 1/4