8. A certain intelligence station has four different passwords $A, B, C, D$, and uses one of them each week. Each week, one of the three passwords not used in the previous week is chosen with equal probability. If the first week uses password $A$, then the probability that the seventh week also uses password $A$ is $\qquad$ (expressed as a simplified fraction).
8. $\frac{61}{243}$.

Let $P_{k}$ represent the probability of using the $A$ type codebook in the $k$-th week. Then the probability of not using the $A$ type codebook in the $k$-th week is $1-P_{k}$.
$$
\begin{array}{l}
\text { Hence } P_{k+1}=\frac{1}{3}\left(1-P_{k}\right)\left(k \in \mathbf{N}_{+}\right) \\
\Rightarrow P_{k+1}-\frac{1}{4}=-\frac{1}{3}\left(P_{k}-\frac{1}{4}\right) \\
\Rightarrow P_{k}-\frac{1}{4}=\frac{3}{4}\left(-\frac{1}{3}\right)^{k-1} \\
\Rightarrow P_{k}=\frac{3}{4}\left(-\frac{1}{3}\right)^{k-1}+\frac{1}{4} \\
\Rightarrow P_{7}=\frac{61}{243} .
\end{array}
$$