圆上 n 点成多边形：包含圆心概率

What is the probability that the center of the circle is inside the polygon

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

A total of $n$ points are chosen uniformly at random on a unit circle, where $n\geq 3$ What is the probability that the center of the circle is inside the polygon formed by these $n$ points?

解析

多边形不包含圆心当且仅当所有点落在同一开半圆内。其概率为 $\frac{n}{2^{n-1}}$ 。

因此包含圆心的概率为

\boxed{1-\frac{n}{2^{\,n-1}}}.