A certain machine used in a manufacturing process can be in one of three states: Fully operational ("full"), partially operational ("part"), or broken ("broken"). If the machine is fully operational today, there is a .7 probability it will be fully operational again tomorrow, a .2 chance it will be partially operational tomorrow, and otherwise tomorrow it will be broken. If the machine is partially operational today, there is a .6 probability it will continue to be partially operational tomorrow and otherwise it will be broken (because the machine is never repaired in its partially operational state). Finally, if the machine is broken today, there is a .8 probability it will be repaired to fully operational status tomorrow; otherwise, it remains broken. Let X = the state of the machine on day n. (a) Identify the state space of this chain. (b) Determine the one-step transition probability matrix. (c) Given the machine is broken today, find the probability that it will stay broken the day after tomorrow.