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 independently of the past, it rains on each trip with probability 0.2. To formulate a Markov chain, let Xn be the number of umbrellas at her current location.

a) Find the transition probability for this Markov chain.
b) Calculate the limiting fraction Stout onowy di of time she gets wet.