Coin sets the number of dice rolls

MediumProbability Conditional Probability & Expectation~5m

Flip a fair coin until the first head; if it lands on flip $n$ , then roll a fair $6$ -sided die $n$ times. What is the probability that the value $1$ appears among those die rolls?

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Given $n$ rolls, $\mathbb{P}(\text{no }1)=(5/6)^n$ . Average over $n$ with weight $(1/2)^n$ .
The sum is geometric in $5/12$ . Don't just use the mean $n=2$ .

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0.2857 (± 0.001)

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Coin sets the number of dice rolls

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