4. Covering a Circle Circumference and a Sphere Surface
Society for Industrial and Applied Mathematics eBooks · 1978
In his turn of the century book Whitworth (1897) was one of the first to consider problems relating to the coverage of a circle by random arcs. Suppose we place n arcs of size a uniformly and independently on the circumference of a circle. Then Whitworth's answer to exercise 667 in his volume can be interpreted as the probability that the part of the circumference that is not covered by an arc consists of exactly n − r connected components, that is, the pattern of arcs leaves n − r gaps on the circumference. He restricted himself, by the comment in exercise 666 “say ” where c is the size of the circumference of the circle, to the case in which .