Generate state matrices by recursion
A school-day record classifies students by their arrival time. The same transition rule is applied repeatedly
Use the state order on time and slightly late. 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 school day. The initial state is S_0, before any applications, so the index counts applications
\[T = \begin{bmatrix} 0.8 & 0.5 \\ 0.2 & 0.5 \end{bmatrix}\]
The initial state column gives counts of students
\[S_0 = \begin{bmatrix} 100 \\ 100 \end{bmatrix}\]
\[S_{n+1} = T S_n\]