10. A certain person goes for a run each morning. When he leaves his house for his run he is equally likely to go out either the front or the back door; and similarly when he returns he is equally likely to go to either the front or back door. The runner owns 5 pairs of running shoes which he takes off after the run at whichever door he happens to be. If there are no shoes at the door from which he leaves to go running he runs barefooted. We are interested in determining the proportion of time that he runs barefooted.

(a) Set this up as a Markov chain. Give the states and the transition probabil-

ities.

(b) Determine the proportion of days that he runs barefooted.