= Solution
For equal rates, the <master equation> generator is $f$ times the symmetric path-graph matrix
$$
W/f=\begin{pmatrix}-1&1&0\\1&-2&1\\0&1&-1\end{pmatrix}.
$$
Its <eigenvectors> may be chosen as $(1,1,1)$, $(1,0,-1)$, and $(1,-2,1)$, with <eigenvalues> $0,-f,-3f$. Resolve the initial empty state as
$$
(1,0,0)=\tfrac13(1,1,1)+\tfrac12(1,0,-1)+\tfrac16(1,-2,1).
$$
Each nonstationary eigencomponent decays with its own rate, giving
$$
\boxed{p_B(t)=\frac13-\frac12e^{-ft}+\frac16e^{-3ft}.}
$$
The remaining <probabilities> are $p_0(t)=1/3+e^{-ft}/2+e^{-3ft}/6$ and $p_A(t)=(1-e^{-3ft})/3$. These satisfy the full equations and sum to one. The limit $p_B(\infty)=1/3$ agrees with the stationary result. At short times $p_B(t)=f^2t^2/2+O(t^3)$: two successive jumps, $0\to A\to B$, are necessary, so a term linear in $t$ would signal an incorrectly included direct transition.
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