Suppose that whether or not it rains today depends on previous weather conditions through the last three days. If it has rained for the past three days, then it will rain today with probability 0.7; if it did not rain for any of the past three days, then it will rain today with probability 0.1; and in any other case the weather today will, with probability 0.6, be the same as the weather yesterday. Explain how this system may be analyzed by using a Markov chain. How many states are needed? Calculate its transition probability matrix.