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Expected gap on 30-sided dice

MediumProbabilityExpected Value~5m

Two fair 3030-sided dice (faces 11–3030) are rolled independently, giving values XX and YY. Compute E[ ∣X−Y∣ ]\mathbb{E}\big[\,|X - Y|\,\big].

Show hints (2)+
  1. There are 2(n−d)2(n-d) ordered pairs at distance dd; sum dd over them.
  2. The closed form is E∣X−Y∣=(n2−1)/(3n)\mathbb{E}|X-Y| = (n^2-1)/(3n) with n=30n=30.

Answer

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9.9889 (± 0.01)

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