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
