We have mentioned that the Markov model in Example 1.2-1 may be too simple for describing the evolution of the weather conditions: in general, it is certainly useful to know the tendency of weather conditions in the past.

However, we can keep using the Markov setup if we extend the set of possible states of the weather on a current day. Namely, we may include in the characterization of states information about the past. For example, if we want to take into account the information about the weather on two consecutive days, we may consider, for a current day, such states as “rainy today, and rainy yesterday,” “rainy today, and normal yesterday,” “icy today, and rainy yesterday,” and so on. Such a model may turn out to be adequate enough.

How many states, should we consider in such a modified model? What is the probability of moving from the state “rainy today, and rainy yesterday” to the state “rainy today, and normal yesterday?”