能站稳吗

这个问题的关键是要认识到，只有当所有 3 个腿不位于同一半圆或相同的 $180^\circ$ 间隔时，桌子才会稳定。要计算此概率，请从圆上的 3 个随机点开始，然后从每个点沿逆时针方向绘制一个半圆。对于任何给定的半圆，其他 2 个点落入其中的概率为 $\frac12\cdot\frac12$。由于所有点不可能位于多个绘制的半圆内，因此这些事件是不相交的，因此我们可以将包含所有点的每个半圆的概率相乘，以获得所有点彼此落在 $180^\circ$ 内的概率。我们取这个值的补集来确定所有三个点都是 $not$ 在同一个半圆内的概率，因此该表成立。我们的答案是： $$1 - 3\cdot\frac14 = \boxed{\frac14}$$

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Original Explanation

The key to this problem is to realize that the table will only be stable if all 3 legs do not lie in a common semicircle, or the same $180^\circ$ interval. To compute this probability, begin with 3 random points on a circle, and draw a semicircle from each point in the counterclockwise direction. For any given semicircle, there will be probability $\frac12\cdot\frac12$ the other 2 points fall into it. Since it is not possible for all the points to lie in more than one of the drawn semi-circles, these events are disjoint and so we can multiply the probabilities of each semi-circle containing all points to obtain the probability all points fall within $180^\circ$ of each other. We take the complement of this value to determine the probability that all three points are $not$ in the same semicircle, and thus the table will stand. Our answer is: $$1 - 3\cdot\frac14 = \boxed{\frac14}$$