An individual possesses 5 umbrellas which he employs in going from his home to the office,...

An individual possesses 5 umbrellas which he employs in going from his home to the office, and vice versa. If he is at home at the beginning of a day and it is raining, then he will take an umbrella with him to the office provided there is one to be taken. Similarly, if he is at the office and at the end of a day it is raining, he will take one to go home (provided there is one to be taken at the office). If it is not raining, then he never takes an umbrella. Assume that, independent of the past, it rains at the beginning or at the end of a day with probability 0.35.

(a)Define a Markov chain for this system by the construction of the one-step transition matrix (Hint: Define the states of the chain as the number of umbrellas the individual has in the place he is at (home or office). Assume that there is a transition each time he changes places (from home to the office or vice versa)

(b)Find the steady state probabilities, by the formulation of the steady state equations.

(c) What fraction of time does the man get wet? Justify your answer.

Expert Solution

Colby Messinger answered 1 year ago

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