# Circle Covering by Arcs

Circle Covering by Arcs

If 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 , and the probability that the arcs leave uncovered gaps is . These results were first proved by L. W. Stevens in 1939. In the image, you can adjust α and 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.

n

P(α,n)=

⌊1/α⌋

∑

k=0

k

(-1)

n |

k |

(n-1)

(1-kα)

l

n |

k |

⌊1/α⌋

∑

j=l

j-l

(-1)

n |

k |

n-1

(1-jα)

n