Expected middle value of three dice

Question

Three fair dice are thrown together. What is the expected value of the middle one of the three results?

Short answer

$\tfrac72$ . Replacing every face $x$ by $7-x$ maps the dice to an equally likely outcome and sends the median $m$ to $7-m$ . So the median's distribution is symmetric about $3.5$ , and its mean is $\tfrac72$ .

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

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Lessons behind this question

Expectation, Mean, and Variance
Counting and Combinatorics
Conditional Expectation and Variance Revisited

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