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Sagot :
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]
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]
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