Diana has regularly contacted three companies (A, B and C) for her quarterly servicing of aircons. If A services her aircons, the probability that there will be no problem with the aircons in the next quarter is 0.8. The same probability for B is 0.7 and for C is 0.6. If all her aircons do not show any problems, she will engage the same company in the next quarter. If any of the aircons shows a problem, then she will engage A with a probability of 0.5, B with a probability of 0.3 or C with a probability of 0.2 in the next quarter. One of her aircons had a problem last quarter.

(i) Identify the probability that her aircons will have no problem in the next quarter.

(ii) Given that her aircons had no problem in the next quarter, what is the probability that she engaged C?

(iii) What is the probability that she will engage A for the next two quarters?