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