Seeing a one when the roll count is geometric

Toss a fair coin until the first head appears; suppose that takes $n$ tosses. Then roll a fair six-sided die $n$ times. What is the probability that at least one of these rolls shows a $1$?

$2/7$. Compute the complement: no $1$ in $n$ rolls has probability $(5/6)^n$, and $n=k$ with probability $2^{-k}$, so $\sum_{k\ge1}(5/12)^k=\tfrac{5/12}{7/12}=\tfrac57$. Hence $1-\tfrac57=\tfrac27$.