The probability that random arcs of angular size cover the circumference of a circle completely (for a circle with unit circumference) is
where is the floor function (Solomon 1978, p. 75). This was first given correctly by Stevens (1939), although partial results were obtains by Whitworth (1897), Baticle (1935), Garwood (1940), Darling (1953), and Shepp (1972).
The probability that arcs leave exactly gaps is given by
Baticle, M. "Le problème des répartitions." Comptes Rendus Acad. Sci. Paris201, 862-864, 1935.Fisher, R. A. "Tests of Significance in Harmonic Analysis." Proc. Roy. Soc. London Ser. A125, 54-59, 1929.Fisher, R. A. "On the Similarity of the Distributions Found for the Test of Significance in Harmonic Analysis, and in Stevens's Problem in Geometric Probability." Eugenics10, 14-17, 1940.Darling, D. A. "On a Class of Problems Related to the Random Division of an Interval." Ann. Math. Stat.24, 239-253, 1953.Garwood, F. "An Application to the Theory of Probability of the Operation of Vehicular-Controlled Traffic Signals." J. Roy. Stat. Soc. Suppl.7, 65-77, 1940.Shepp, L. A. "Covering the Circle with Random Arcs." Israel J. Math.11, 328-345, 1972.Siegel, A. F. Random Coverage Problems in Geometric Probability with an Application to Time Series Analysis. Ph.D. thesis. Stanford, CA: Stanford University, 1977.Solomon, H. "Covering a Circle Circumference and a Sphere Surface." Ch. 4 in Geometric Probability. Philadelphia, PA: SIAM, pp. 75-96, 1978.Stevens, W. L. "Solution to a Geometrical Problem in Probability." Ann. Eugenics9, 315-320, 1939.Whitworth, W. A. DCC Exercises in Choice and Chance. 1897. Reprinted New York: Hafner, 1965.
that 
random arcs of angular size 
cover the circumference of a circle completely (for a circle with unit circumference) is 
is the floor function (Solomon 1978, p. 75). This was first given correctly by Stevens (1939), although partial results were obtains by Whitworth (1897), Baticle (1935), Garwood (1940), Darling (1953), and Shepp (1972). 
arcs leave exactly 
gaps is given by 
