Solution
= Solution
The states $b,d$ form a <closed communicating class>. Set $p_n=\mathbb P(X_n=b\mid X_0=d)$. Within this class,
$$
p_{n+1}=\frac14p_n+\frac23(1-p_n)=\frac23-\frac5{12}p_n,\qquad p_0=0.
$$
Its fixed point is $8/17$. Subtracting that fixed point and iterating gives \b[the exact transition probability]
$$
\boxed{p_n=\frac8{17}\left(1-\left(-\frac5{12}\right)^n\right),\qquad n\ge0.}
$$