To find the probability of transitioning from state [tex]\( A \)[/tex] to state [tex]\( B \)[/tex] in two trials, we need to look at the appropriate entry in the matrix [tex]\( P^2 \)[/tex], which represents the probability of transitioning between states in two steps.
Given the transition matrix [tex]\( P \)[/tex] and its second power [tex]\( P^2 \)[/tex]:
[tex]\[
P = \begin{bmatrix}
A & B & C \\
0 & 0.3 & 0.7 \\
0.2 & 0.2 & 0.6 \\
0.1 & 0 & 0.9
\end{bmatrix}
\][/tex]
[tex]\[
P^2 = \begin{bmatrix}
A & B & C \\
0.13 & 0.06 & 0.81 \\
0.10 & 0.10 & 0.80 \\
0.09 & 0.03 & 0.88
\end{bmatrix}
\][/tex]
The entry in the second row and second column of [tex]\( P^2 \)[/tex] (0.06), represents the probability of transitioning from state [tex]\( A \)[/tex] to state [tex]\( B \)[/tex] in exactly two steps.
Hence, the probability of going from state [tex]\( A \)[/tex] to state [tex]\( B \)[/tex] in two trials is:
[tex]\[
\boxed{0.06}
\][/tex]
