For Example 12.10, where a mouse is moving inside the given maze, find the probability that the mouse is in cell i, 1 ≤ i ≤ 9, at a random time in the future. Example 12.10 Suppose that a mouse is...

For Example 12.10, where a mouse is moving inside the given maze, find the probability that the mouse is in cell i, 1 ≤ i ≤ 9, at a random time in the future.

Example 12.10

Suppose that a mouse is moving inside the maze shown in Figure 12.1, from one cell to another, in search of food. When at a cell, the mouse will move to one of the adjoining cells randomly. For n ≥ 0, let X_n
be the cell number the mouse will visit after having changed cells n times. Then {X_n
: n = 0, 1,...} is a Markov chain with state space {1, 2, . . . , 9} and transition probability matrix