Ticket price for a one-roll die payout

Question

A single fair six-sided die is rolled and you receive, in dollars, the number that comes up. What entry fee would make this a fair game?

Short answer

$3.50. A fair price equals the expected payout, and the faces $1$ to $6$ are equally likely, so the mean is $\tfrac{1+2+\dots+6}{6}=3.5$ .

Adapted from Crack, Heard on the Street, Q4.2, p.37; 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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