Chapter 8.3, Problem 23E

Mouse in a Maze A mouse is placed in one of the compartments of the maze shown in Fig. 7. After each minute of looking unsuccessfully for food, the mouse exits through one of the doors at random and moves to an adjacent compartment. The Exit door is one way; that is, the mouse cannot return after it has exited the maze.

Figure 7

a. Draw the transition diagram for the Markov process.

b. Set up an absorbing stochastic matrix for the Markov process.

c. Find the stable matrix.

d. If the mouse begins in compartment A, what is the expected amount of time that it spends in the maze?

Note: ${\left[\begin{array}{ccc}1& -\frac{1}{2}& -\frac{1}{2}\\ -\frac{2}{3}& 1& -\frac{1}{2}\\ -\frac{1}{3}& -\frac{1}{4}& 1\end{array}\right]}^{-1}=\left[\begin{array}{ccc}4.2& 3& 3.6\\ 4& 4& 4\\ 2.4& 2& 3.2\end{array}\right]$.