= Solution
Let $h$ be the probability of ever hitting $c$ starting at $a$. A <first-step analysis> gives a direct success with probability $1/3$, a return to the same problem with probability $1/3$, and entry into the closed class $\{b,d\}$ with probability $1/3$, after which $c$ cannot be reached. Therefore
$$
h=\frac13+\frac13h,
\qquad\boxed{h=\frac12.}
$$
The possibility of infinitely many self-loops has probability zero and introduces no extra solution of this equation.
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