Determine the long-run state of a transition model
A weekly program records each participant's exercise status. The same transition proportions apply in every period
One application of the transition matrix represents one week. In \(S_n\), the index \(n\) counts the number of applications from \(S_0\)
Use the state order Target met and Partly met. Each column of \(T\) is a current state and each row is the next state. The entries of each state matrix are counts of participants
\[T = \begin{bmatrix} 0.75 & 0.4 \\ 0.25 & 0.6 \end{bmatrix}\]
\[S_0 = \begin{bmatrix} 25 \\ 261 \end{bmatrix}\]
Use probe indices 1 and 6. Give probe counts to the nearest whole count