三脚桌能站稳的概率

桌子站稳当且仅当圆盘的质心（圆心）落在三条桌腿形成的三角形内。

等价于“圆周上三点形成的三角形包含圆心”的概率，结果为

$$\frac{1}{4}.$$

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Original Explanation

Let's fix the first leg at a random spot on the circumference. The second leg is expected to be $\frac{1}{4}^{\text{th}}$ of the circumference away from the initial leg (draw out the circle, you will see that the average spot for the second leg is half way between where the first leg is and the opposite point of the first leg). Finally, for the table to stand up, the center of the table (center of mass) needs to be within the triangle that is formed by connecting the three legs. This leaves us with $\frac{1}{4}^{\text{th}}$ of the circumference to place the final leg which allows for the table to stand. Thus the answer is $\frac{1}{4}$.