Circle Covering by Arcs
If points are chosen at random on a circle with unit circumference, and an arc of length α is extended counterclockwise from each point, then the probability that the entire circle is covered is , and the probability that the arcs leave uncovered gaps is . These results were first proved by L. W. Stevens in 1939. In the image, you can adjust α and and compare observed circle coverings to the theory. Note that, especially when the arc length is small, there is a reasonable chance that some of the uncovered gaps will be too small to see.