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Weird probability distributions on dyadic rationals from a simple averaging process (mathoverflow.net/q/387543/440): start with the two-element multiset {0,1}, repeatedly draw two-element samples (without replacement) from the multiset and include the average into the multiset. The result tends to cluster around some value, not the same value every time but itself drawn from a bimodal distribution, with the clustering sort of looking Cauchy but not quite. Interesting new MathOverflow question.

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