Let S and C denote a Sunny day and a Cloudy day, respectively. Suppose that the weather on the given day depends on the past weather only through the weather on the two immediately preceding days according to the following conditional probabilities: The probability that a day is sunny given that the immediately preceding two days were sunny is P(S|S,S) = .8. The probability that a day is sunny given that the immediately preceding day was sunny and the day before was cloudy is P(S|C,S) = .6.The probability that a day is sunny given that the immediately preceding day was cloudy and the day before was sunny is P(S|S,C) = .4. Similarly, P(S|C,C) = .1, P=(C|S,S) =.2, P(C|C,S) = .4, P(C|S,C) = .6, P(C|C,C) = .9. Let n = 0 denote the initial day (day 0), let n = 1 denote the 1st day, and so on. We also know the following two initial probabilities: Prob(the initial day is cloudy, the first day is sunny) = .1 and Prob (the initial day is cloudy, the first day is cloudy) = .9.
a.) What is the probability that the second and third day will be cloudy?
b.) What is the probability that the third day will be cloudy?
c.) What is the long-term probability that a day will be cloudy?