Solution
= Solution
{c}
Condition on the intermediate state. Starting at $+1$, a return to $+1$ after two transitions occurs through $-1$ with probability $q(1-p)$ or through $+1$ with probability $(1-q)^2$. These alternatives are disjoint, so
$$
\boxed{\mathbb P(X_3=+1\mid X_1=+1)=q(1-p)+(1-q)^2.}
$$