In Example 11.4, let a = 0 and b = 1=2. Find P; \(P^{2}\); and \(P^{3}\): What would \(P^{n}\) be? What happens to \(P^{n}\) as n tends to infinity? Interpret this result.
The initial state represents the President’s choice. \[ \begin{matrix} & yes \\ & no \\ \end{matrix}\begin{pmatrix} 1-a & a \\ b & 1-b \\ \end{pmatrix} \]
The initial state \(P^{1}\) is given as:
\[P^{1} = \begin{bmatrix} 1 & 0 \\ 0.5 & 0.5 \\ \end{bmatrix}\]
\[P^{2} = P^{1}P^{1} = \begin{bmatrix} 1 & 0 \\ 0.5 & 0.5 \\ \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0.5 & 0.5 \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0.75 & 0.25 \\ \end{bmatrix}\]
\[P^{3} = P^{2}P^{1} = \begin{bmatrix} 1 & 0 \\ 0.75 & 0.25 \\ \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0.5 & 0.5 \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0.875 & 0.125 \\ \end{bmatrix}\]
\[P^{4} = P^{3}P^{1} = \begin{bmatrix} 1 & 0 \\ 0.875 & 0.125 \\ \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0.5 & 0.5 \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0.9375 & 0.0625 \\ \end{bmatrix}\]
\[P^{n} = \begin{bmatrix} 1 & 0 \\ 2n-1/2^n & 0.5^n \\ \end{bmatrix} \]
As \(P^{n}\) tend to infinity, \(P_{21}\) tend to 1 while \(P_{22}\) tends to zero.