A professor living in a tropical climate has 3 umbrellas. He goes between home and work, taking an umbrella if one is available but only if it is raining. Being a tropical climate, the probability it is raining in the morning when he goes to campus is 0.15 but when he leaves to go home in the afternoon, the probability it is raining is 0.65. (a) Describe a Markov Chain that models this situation. (b) After many days of this behavior, what is the probability there is no umbrella when the professor is at work? (c) After many days of this behavior, what is the probability there is no umbrella when the professor is at home? (d) After many days of this behavior, what is the probability the professor will get wet when going to work from home? (e) One day, the professor brings all umbrellas home but then he resumes the same behavior described above. What is the expected number of days until there is no umbrella at home? (f) Frustrated, the professor decides to always take an umbrella to work from home if one is available at home when he leaves but sticks to the 01d strategy when leaving work. Following this new strategy, what is the probability he has to commute (in either direction) without an umbrella when it is raining