Four fives before the first six

MediumProbability Random Variables~5m

You roll a fair $6$ -sided die until you get your first $6$ . What is the probability that a $5$ appears exactly four times before that first $6$ ?

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Rolls of $1,2,3,4$ don't matter - restrict to $\{5,6\}$ rolls, each equally likely.
Then the count of $5$ s before the first $6$ is geometric: $\mathbb{P}(k)=(1/2)^{k+1}$ .

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0.03125 (± 0.0005)

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Four fives before the first six

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