Mean gap between two thirty-sided dice

Question

Two fair dice, each numbered $1$ to $30$ , are rolled independently, giving $X$ and $Y$ . Compute $\mathbb{E}\,|X-Y|$ .

Short answer

$\tfrac{899}{90}$ . For $k\ge1$ there are $2(30-k)$ outcomes with $|X-Y|=k$ , so the mean is $\tfrac{1}{900}\sum_{k=1}^{29}2k(30-k)=\tfrac{8990}{900}$ . In general an $n$ -sided die gives $\tfrac{n^2-1}{3n}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Expectation, Mean, and Variance
Independence

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