Dice game paying the first non-seven total

Question

Two fair dice are thrown repeatedly until their total is something other than $7$ ; you are then paid that total. What is a fair price for this game?

Short answer

$7$ . The payout is the dice total conditioned on not being 7. The distribution of the total is symmetric about 7 (e.g. 2 and 12 are equally likely), and removing 7 keeps that symmetry, so the conditional mean is still 7.

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Lessons behind this question

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

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