Problem 1. A manufacturer has a machine that, when operational at the beginning of a day, has a probability of 0.1 of breaking down sometime during the day. When this happens, the repair is done the next day and completed at the end of that day. (a) Formulate the evolution of the status of the machine as a Markov chain by identifying three possible states at the end of each day, and then constructing the (one-step) transition matrix. (b) Find the probability that the machine will be operational for 5 straight days before breaking down. Compare this with the probability that a currently operational ma- chine will be broken down on day 6. (c) Find the expected first passage time ,, from state i to state j, for all i and j. Use these results to identify the expected number of full days that the machine will remain operational before the next breakdown after a repair is completed.