With these three ingredients, you can calculate all the probabilities you need. Suppose that right now, your probability that the weather will be dry tomorrow is 0.78. Of course, this Markov chain is too simple to tell us anything useful about the weather. A much more sophisticated state space would be needed for that. Perhaps, even, we would have to make the transition probabilities change as time went by.
P is the standard symbol for a matrix that contains all the transition probabilities.
This matrix is called the one-step transition probability matrix . Each row is a probability distribution for a given current state to a possible next state.