\begin{pmatrix}
P_A(i+1) \\
P_B(i+1) \\
\end{pmatrix}
=
\begin{pmatrix}
1/2 & 1 \\
1/2 & 0 \\
\end{pmatrix}
\begin{pmatrix}
P_A(i) \\
P_B(i) \\
\end{pmatrix}
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