Generate state matrices by recursion
A weekly team record classifies each player's availability. The same transition rule is applied repeatedly
Use the state order available and limited. Every state matrix is a column in this order. In the transition matrix, columns are current states and rows are next states
One application of the transition matrix represents one week. The initial state is \(S_0\), before any applications, so the index counts applications
\[T = \begin{bmatrix} 0.8 & 0.3 \\ 0.2 & 0.7 \end{bmatrix}\]
The initial state column gives counts of players
\[S_0 = \begin{bmatrix} 100 \\ 100 \end{bmatrix}\]
\[S_{n+1} = T S_n\]