圆内矩形：以 PQ 为对角线的概率

Rectangle is inside the circle

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Consider a random point $P$ on the circumference of the unit circle centered at $(0,0)$ and a random point $Q$ inside the circle. Using $PQ$ as diagonal, a rectangle is drawn with sides parallel to $x$ - and $y$ - axis. What is the probability that the rectangle is inside the circle?

解析

设单位圆上点 $P=(\\cos\\alpha,\\sin\\alpha)$ ，圆内均匀点 $Q$ 。以 $PQ$ 为对角线且边平行坐标轴时，矩形在圆内当且仅当 $Q$ 落在以 $P$ 与其关于坐标轴对称点构成的矩形内。\n\n该矩形面积为 $4\\sin\\alpha\\cos\\alpha$ ，圆面积为 $\\pi$ ，故\n\n $\n\\mathbb{P}(\\text{满足}\\mid\\alpha)=\\frac{4\\sin\\alpha\\cos\\alpha}{\\pi}.\n$ \n\n对 $\\alpha$ 平均可得\n\n $\n\\mathbb{P}=\\boxed{\\frac{4}{\\pi^2}}.\n$