 # Let n points be placed uniformly at random on the boundary of a circle of circumference 1. Micah Haynes 2022-05-09 Answered
Let $n$ points be placed uniformly at random on the boundary of a circle of circumference 1.

These $n$ points divide the circle into $n$ arcs.

Let ${Z}_{i}$ for $1\le i\le n$ be the length of these arcs in some arbitrary order, and let $X$ be the number of ${Z}_{i}$ that are at least $\frac{1}{n}$.

What is $E\left[X\right]$ and $Var\left[X\right]$?

Any hints will be appreciated. Thanks..

(By the way this problem is exercise 8.12 from the book 'Probability and Computing' by Mitzenmacher and Upfal)
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If you cut the circle along the first placed point, you can see that the situation is equivalent to taking the interval [0,1] and placing $n-1$ points uniformly at random into the interval.