An individual has three umbrellas; some at her office and some at home. If she is leaving home in the morning (or leaving work at night) and it is raining, she will take an umbrella, if one is there. Otherwise, she 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 her current location.

(a) Find the transition probability for this Markov chain.

(b) Calculate the limiting fraction of time she gets wet.