Exactly four fives before the first six

Question

James rolls a fair die repeatedly and stops at his first $6$ . What is the probability that a $5$ appeared exactly four times among his rolls?

Short answer

$\tfrac1{32}$ . Ignore every roll other than $5$ or $6$ ; each relevant roll is a $5$ or a $6$ with probability $\tfrac12$ each. We need the relevant sequence to read $5,5,5,5,6$ , which has probability $2^{-5}=\tfrac1{32}$ .

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Lessons behind this question

Counting and Combinatorics
Expectation, Mean, and Variance
Conditional Probability, Total Probability, and Bayes' Rule

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