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

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Advance Math

An individual possesses 5 umbrellas which he employs in goingfrom his home to the office, and vice versa. If he is at home atthe beginning of a day and it is raining, then he will take anumbrella 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 israining, he will take one to go home (provided there is one to betaken at the office). If it is not raining, then he never takes anumbrella. Assume that, independent of the past, it rains at thebeginning or at the end of a day with probability 0.35.

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

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

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

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