Rolling a 5-sided die to reach five

MediumProbability Expected Value~7m

A fair $5$ -sided die (faces $1$ through $5$ ) is rolled repeatedly until the running total is at least $5$ . What is the expected number of rolls? Round to four decimal places.

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Let $e_s$ be expected extra rolls from total $s$ ; $e_s = 1 + \tfrac15\sum_{r=1}^5 e_{s+r}$ with $e_s=0$ for $s\ge5$ .
Solve backward from $e_4=1$ . You'll find the clean pattern $e_0=(6/5)^4$ .

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2.0736 (± 0.001)

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Rolling a 5-sided die to reach five

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