Problem 1 [9 points) a. [4 points) Suppose that on any given day, a machine can be in good condition (G), in poor condition (P), or broken (B). If the machine is in good condition on a particular day, then it remains in good condition on the next day with probability 0.7, it is in poor condition the next day with probability 0.2, and with probability 0.1 it is broken the next day. Similarly, if the machine is in poor condition on a particular day, then it remains in poor condition on the next day with probability 0.6, and it is broken on the next day with probability 0.4. Finally, with probability 0.5, it takes exactly one day to repair a broken machine; otherwise (i.e., with probability 0.5), it takes exactly two days to repair a broken machine. Upon repair completion, the machine is in good condition. Show that this situation can be modeled using a Markov chain (i.e., specify your model, show it is a Markov chain, determine its state space, provide its transition probability matrix, and state its initial distribution if appropriate information is provided}. Explain. b. (5 points) Suppose that a given product is sold in containers of four units per box, and that we have two boxes (or eight units) at our disposal. Suppose furthermore that six of the available units are in working condition, with the remaining two units being defective. Consider a process in which one repeatedly selects a unit at random from each box, and then (simultaneously) places the unit drawn from the rst box in the second box, and conversely with the unit drawn from the second box. Assume that the four units that are initially in the rst box were chosen at random. Model this process using a Markov chain. Make sure that your model can be used to keep track of how many goodfdefective units are in each box at any given time. Explain