Mr. Mean Variance has the only key which locks or unlocks the door to Building 59, the Probability Building. He visits the door once each hour on the hour. When he arrives: If the door is unlocked, he locks it with probability 0.3. If the door is locked, he unlocks it with probability 0.8. (a) After he has been on the job several months, is he more likely to lock the door or to unlock it on a randomly selected visit? (b) With the process in the steady state, Joe arrived at Building 59 two hours ahead of Harry. What is the probability that each of them found the door in the same condition? (c) Given the door was unlocked at the time Mr. Variance was hired, determine the expected value of the number of visits up to and including the one on which he unlocks the door himself for the first time. In the following, suppose that every time after he leaves the building, he comes back after T hours, where T is equal to 1 with probability 1/2, and to 2 with probability 1/2. (d) After he has been on the job several months, an absent-minded visitor, Mr. Drunker, visits Building 59 five minutes before a randomly-selected hour. What is the probability that Mr. Drunker will find the door is unlocked? (Hint: Consider modeling the problem as a new Markov process.) (e) With the process in the steady state, Joe arrives at Building 59 two hours ahead of Harry. What is the probability that each of them found the door in the same condition?