Rolls until a running total divisible by six

Question

A fair six-sided die is rolled again and again while you keep a running total of the results. You stop as soon as that total is divisible by 6. On average, how many rolls does this take?

Short answer

$6$ . Whatever the current total is modulo 6, exactly one face brings it to a multiple of 6, so every roll ends the game with probability $\tfrac16$ independently. The roll count is geometric with mean 6.

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Lessons behind this question

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

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