Expected rolls to see a 5 followed by a 6

Question

A fair die is rolled until a $5$ is immediately followed by a $6$ . What is the expected number of rolls?

Short answer

$36$ . Track whether the last roll was a $5$ . From scratch, $E_0=1+\tfrac16E_1+\tfrac56E_0$ ; after a $5$ , a $6$ finishes, another $5$ keeps the state and anything else resets: $E_1=1+\tfrac16E_1+\tfrac46E_0$ . Solving gives $E_0=36$ (the pattern $56$ has no self-overlap, so the answer is simply $6^2$ ).

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

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Lessons behind this question

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

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