An individual has three umbrellas, some at his office, and some at home. If he is leaving home in the morning (or leaving work at night) and it is raining, he will take an umbrella, if there is one there. Otherwise, he gets wet.

Assume that, independent of the past, it rains on each trip with probability 0.2. To formulate a Markov chain, let X_n be the number of umbrellas at his current location "before" he starts his n-th trip. Note that "current location" can either be home or office, depending on whether the trip is from home to office or vice versa.

a. Find the transition probabilities of this Markov chain.

b. Calculate the limiting fraction of time he gets wet.