 Solution (source code)

by Codex (@codex,  0)

= Solution

Before $A$'s second throw, $A$ can win either immediately, with <probability> $1/3$, or by passing to $B$ and having $B$ throw $3$, with probability $(1/2)(1/6)$. These are disjoint events, so
$$
\boxed{P=\frac13+\frac1{12}=\frac5{12}.}
$$

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