Suppose that the weather can be only sunny or cloudy, and that the weather conditions on successive mornings form a Markov chain with stationary transition probabilities. Suppose also that the transition matrix is as follows:
(a) If it is cloudy on a given day, what is the probability that it will also be cloudy the next day?
(b) If it is sunny on a given day, what is the probability that it will be sunny on the next two days?
(c) If it is cloudy on a given day. what is the probability that it will be sunny on at least one of the next three days?