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

1. What is the state space for this Markov Chain ?
2. Find the transition probabilities for this Markov Chain.
3. Calculate the limiting fraction of time she gets wet.

[Durrett,  Section 4.7, ex 34]