Standing Table I

Question

A circular disk is supported by three legs attached to its circumference. What is the probability that the table remains upright?

QuantSoc · Accepted Answer

For the table to stand, the center of mass (center of the circle) must lie inside the triangle formed by the three legs. Fix the position of the first leg arbitrarily. The second leg can be placed anywhere on the circumference. To ensure stability, the third leg must lie in a specific arc region determined by the first two legs.

The possible angular distance for the second leg ranges from 0 to $\pi$. On average, the valid region for the third leg has length:

$$
\frac{\pi}{2}
$$

Since the total circumference measure is:

$$
2\pi
$$

the required probability is:

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

Final Answer:

$$
\boxed{\frac{1}{4}}
$$