方形内两点成直径：圆心落在圆内概率
The circle formed by taking the two points as diameter
题目详情
What is the probability that two uniform random points in a square are such that the center of the square lies inside the circle formed by taking the two points as diameter?
解析
圆心 O 在以 AB 为直径的圆内当且仅当 ∠AOB > 90°。固定 A，过 O 作垂直于 OA 的直线 l，则 B 在 l 的另一侧当且仅当 ∠AOB > 90°。该直线把正方形分成两块等面积区域，因此概率为