\begin{pmatrix}
P_A(i) \\
P_B(i) \\
\end{pmatrix}
=
\begin{pmatrix}
\frac{2}{3} - \frac{2 \left(- \frac{1}{2}\right)^{i}}{3}\\
\frac{2 \left(- \frac{1}{2}\right)^{i}}{3} + \frac{1}{3}
\end{pmatrix}
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