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The riddle of the week is rather standard probability calculus

If N points are generated at random places on the perimeter of a circle, what is the probability that you can pick a diameter such that all of those points are on only one side of the newly halved circle?

Since it is equivalent to finding the range of N Uniform variates less than ½. And since the range of N Uniform variates is distributed as a Be(N-1,2) random variate. The resulting probability, which happens to be exactly $N/2^{N-1}$, is decreasing exponentially, as shown below…