Fives thrown before the first six

Question

A fair die is thrown over and over until a $6$ first appears. On average, how many $5$ s are thrown before that happens?

Short answer

$1$ . Only 5s and 6s matter: each throw that is a 5 or 6 is equally likely to be either. So the number of 5s before the first 6 is geometric with success probability $\tfrac12$ (counting failures), whose mean is $1$ .

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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