Rolls needed for back-to-back sixes

Question

A fair die is rolled until two $6$ s appear on consecutive rolls. What is the expected number of rolls?

Short answer

$42$ . Let $E_0$ be the expected remaining rolls from scratch and $E_1$ after a single 6. Then $E_1=1+\tfrac56E_0$ and $E_0=1+\tfrac16E_1+\tfrac56E_0$ . Solving gives $E_0=42$ (the general formula $p^{-1}+p^{-2}$ with $p=\tfrac16$ ).

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 Probability, Total Probability, and Bayes' Rule

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