A mole's dwelling consists of 4 small caves (numbered 1 through 4) and several tunnels connecting them, as seen in the figure. The mole always sleeps overnight in a cave, but may move during the day to another cave (not necessarily adjacent). A long term observation reveals the following habits. It the mole sleeps in cave 1 during one night, then the following night he sleeps in cave 4 with probability 0.1, but is equally likely to sleep in each of the three other caves. It the mole sleeps in cave 2 during one night, then the following night he sleeps in cave 2 with probability 0.1, in cave 4 with probability 0.1, and is equally likely to sleep in cave 1 as in cave 3. It the mole sleeps in cave 3 during one night, then the following night he will definitely not sleep in cave 4, will sleep in cave 3 with probability 0.1, and is twice as likely to sleep in cave 1 as in cave 2. It the mole sleeps in cave 4 during one night, then the following night he sleeps in the same cave with probability 0.4, in cave 1 with probability 0.3, and in cave 2 with probability 0.2. 4 2 (a) Set up the transition matrix for this Markov chain. (b) If the mole sleeps in cave 3 tonight, what is the probability that he will sleep in cave 4 three nights later? (c) If tonight the mole sleeps in caves 1, 2, 3, and 4 with probabilities 0.3, 0.2, 0.1, and 0.4, respectively, what is the probability that he will sleep in cave 4 three nights later? (d) How do we know this Markov chain possesses a steady-state vector? (e) In the long run, what percentage of time does the mole sleep in each of the four caves?