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Circle Covering by Arcs

​
arc length α
0.1
number of arcs n
5
random seed
If
n
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
P(α,n)=
⌊1/α⌋
∑
k=0
k
(-1)
n
k
(n-1)
(1-kα)
, and the probability that the arcs leave
l
uncovered gaps is
n
k
⌊1/α⌋
∑
j=l
j-l
(-1)
n
k
n-1
(1-jα)
. These results were first proved by L. W. Stevens in 1939. In the image, you can adjust α and
n
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.
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