5. On a given day, Emmett drives to work (state 1), takes the train (state 2), or hails a taxi

(state 3). Let Xn = 1 if he drives to work on day n, Xn = 2 if he takes the train on day n, and Xn = 3 if he hails a taxi on that day. Suppose that {Xn : n = 1, 2, . . .} is a Markov chain, and depending on how Emmett went to work the previous day, the probability of choosing any one of the means of transportation is given by the following transition probability matrix:

(a) Given that Emmett took the train today and every day in the last five days, what is the probability that he will not take the train to work tomorrow?

(b) If Emmett took the train to work today, what is the probability that he will not take the train to work tomorrow and the day after tomorrow?

(1/6 2/3 1/6 P=1/2 P 1/2 1/3 1/6 2/5 1/2 1/10/