Name:___________________________________ Date: ____________

1. Each semester every student is assigned to one of three states. The student has normal status (N) if all is well. The student is assigned probationary status (P) if the previous semester's grades are poor. The student is suspended (S) for a semester if grades are not improved while on probation. We assume that students always return to normal status after a semester of being suspended. The probabilities observed are N-N = 2/3, N-P = 1/3, P-N = 1/2, P-S = 1/2, and S-N = 1.

        a. Draw directed graph for the relation M on three vertices N, P, and S
            using the probabilities listed for each state change.



 

        b. Find the in-degree and out-degree of each vertex in M.
            List any sources or sinks.

        c. Fill in the weighted adjacency matrix, M, representing this Markov chain.

        d. Find M².




        e. What is the chance that a student with normal status will be in S (suspended status)
            after two semesters?
                                                                                                                                                                                                                                 

        f. Given M^∞, what is the long term probability that a student will
            achieve N status?  ...S status?