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Find the probability of going from state [tex]$A$[/tex] to state [tex]$B$[/tex] in two trials, given the transition matrix [tex]$P$[/tex] and the second power of [tex]$P$[/tex] below.

[tex]\[ P = \begin{bmatrix}
A & B & C \\
0 & 0.3 & 0.7 \\
0.2 & 0.2 & 0.6 \\
0.1 & 0 & 0.9
\end{bmatrix} \quad \text{and} \quad P^2 = \begin{bmatrix}
A & B & C \\
0.13 & 0.06 & 0.81 \\
0.10 & 0.10 & 0.80 \\
0.09 & 0.03 & 0.88
\end{bmatrix} \][/tex]

The probability of going from state [tex]$A$[/tex] to state [tex]$B$[/tex] in two trials is [tex]$\square$[/tex].

Sagot :

GinnyAnswer
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]
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