A production line comprises three machines working independently of each other. For each machine, the probability of working through the day is p ∈ (0, 1) and may breakdown during a day with probability q = 1 − p, independently of its previous history. There is a single repairman who, if he has to, repairs exactly one machine overnight. If there is a machine not working by the end of the day, then the repairman works during the night. Let Xn denote the number of working machines at the end of day before the repairman begins any over-night repair. a-) Specify the state space of (Xn : n ≥ 0).

b-) Determine the transition matrix in terms of p.

c-) Given that 2 machines are idle tonight, what is the probability of one idle machine 3 nights later, if p = 0.8 (answer 6 decimal places).