Generate state matrices by recursion
A daily record classifies the weather at a group of locations. The same transition rule is applied repeatedly
Use the state order dry and overcast. 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 day. The initial state is S_0, before any applications, so the index counts applications
\[T = \begin{bmatrix} 0.8 & 0.2 \\ 0.2 & 0.8 \end{bmatrix}\]
The initial state column gives counts of locations
\[S_0 = \begin{bmatrix} 100 \\ 150 \end{bmatrix}\]
\[S_{n+1} = T S_n\]