Solution
= Solution
The <Markov property> and conditioning on the intermediate state give the bridge probability
$$
\mathbb P(X_t=2\mid X_0=1,X_{3t}=1)
=\frac{p_{12}(t)p_{21}(2t)}{p_{11}(3t)}.
$$
Substitution from part (i) yields
$$
\boxed{
\frac{\lambda\mu(1-e^{-st})(1-e^{-2st})}
{s(\lambda+\mu e^{-3st})}}
\qquad(s=\lambda+\mu).
$$