圆周三点三角形包含圆心的概率

经典结论：三点所成三角形包含圆心，当且仅当这三点不落在同一个半圆内。

而三点落在同一半圆内的概率为 $3/4$，因此包含圆心的概率为

$$1-\frac{3}{4}=\frac{1}{4}.$$

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Original Explanation

Fix the first point arbitrarily. The second point can also lie anywhere, but notice that to form a triangle which contains the center, the third point must lie on the portion which has equivalent size as the length of the arc between the first two points, reflected over the center. Thus, think about the position of the second point. The size of that portion could be anywhere between $0$ and $\pi$. On average, it is $\dfrac{\pi}{2}$, and hence the answer is

$$\dfrac{\frac{\pi}{2}}{2\pi} = \dfrac{1}{4}$$