The probability P(a, n) that n random arcs of angular size a cover the circumference of a circle completely (for a circle with unit circumference) is P(a, n) = sum_(k = 0)^(⌊1/a⌋) (-1)^k(n k)(1 - k a)^(n - 1), where ⌊x⌋ is the floor function. This was first given correctly by Stevens, although partial results were obtains by Whitworth, Baticle, Garwood, Darling, and Shepp. The probability that n arcs leave exactly l gaps is given by P_(l gaps)(a, n) = (n l) sum_(j = l)^(⌊1/a⌋) (-1)^(j - l)(n - l j - l)(1 - j a)^(n - 1) (Stevens 1939; Solomon 1978, pp. 75-76).

circle line picking | circle point picking