Solution
= Solution
Write $q=35/36$. Alice wins on her $(n+1)$st turn if $n$ complete rounds fail and then she succeeds. These disjoint events give the <geometric series>
$$
P_A=\sum_{n=0}^{\infty}q^{2n}p=\frac{p}{1-q^2}
=\frac{1}{1+q}.
$$
Consequently
$$
\boxed{P_A=\frac{36}{71}}.
$$
The <probability> of never obtaining a winner is $\lim_{n\to\infty}q^{2n}=0$. The small advantage over $1/2$ comes from taking the first turn.