= Solution
This is a <two-turn renewal game>. A full pair of throws has probability $(1/2)^2=1/4$ of passing twice and returning to the original state. The probability that $A$ wins before that restart is $5/12$. Thus <first-step analysis> gives $p=5/12+p/4$, or
$$
\boxed{P(A\text{ wins})=\frac59.}
$$
Equivalently sum the <geometric series> $(5/12)\sum_{j\geq0}(1/4)^j$. The probability of surviving infinitely many pairs is zero, so the eventual winner is well defined with probability one.
Back to article page
