Suppose that there are two doctors in a country town, Dr Black and Dr White. Each year, 13% of patients move from Dr Black to Dr White, while 19% of patients move from Dr White to Dr Black. Suppose that initially 33% of patients go to Dr Black and 67% go to Dr White. Find the percentage of patients who will eventually be attending each doctor if this pattern continues indefinitely.
Ok, so I said the transition matrix $T= \left[ {\begin{array}{cc} 0.87 & 0.19 \\ 0.13 & 0.81 \\ \end{array} } \right] $ And that led me to say that eventually 59.4% of patients will go to Dr Black and 40.6% will go to Dr. White. But my textbook disagrees. Is my transition Matrix wrong. Any help would be much appreciated.