Chapter 8.1, Problem 25E

T-Maze Each day, mice are put into a T-maze (a maze shaped like a “T”; Fig. 7). In this maze, they have the choice of turning to the left (are rewarded with cheese) or to the right (receive cheese along with mild shock). After the first day, their decision whether to turn left or right is influenced by what happened on the previous day. Of those that go to the left on a certain day, 90% go to the left on the next day and 10% go to the right. Of those that go to the right on a certain day, 70% go to the left on the next day and 30% go to the right.

Figure 7

a. Set up the $2\times 2$ stochastic matrix with columns and rows labeled L and R that describes this situation.

b. Compute the second power of the matrix in part (a).

c. Suppose that, on the first day (day 0), 50% go to the left and 50% go to the right. So, the initial distribution is given by the column matrix ${\left[\begin{array}{l}.5\\ .5\end{array}\right]}_0$. Using the matrices in parts (a) and (b), find the distribution matrices for the next two days, days 1 and 2.

d. Make a guess as to the percentage of mice that will go to the left after 50 days. (Do not compute.)