Solution
= Solution
By stationarity,
$$
\begin{aligned}
\mathbb P(X_{n-2}=B\mid X_n=B)
&=\frac{\mathbb P(X_{n-2}=B,X_n=B)}{\mathbb P(X_n=B)}\\
&=\frac{\pi_B(P^2)_{BB}}{\pi_B}
=(P^2)_{BB}.
\end{aligned}
$$
The two-step return probability is
$$
(P^2)_{BB}
=P_{BA}P_{AB}+P_{BB}^2+P_{BC}P_{CB}
=\frac34\frac14+0+\frac14\frac18.
$$
Therefore the requested probability is
$$
\boxed{\frac7{32}}.
$$