 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. ThereforeThe possibility of infinitely many self-loops has probability zero and introduces no extra solution of this equation.