Suppose that the weather in a particular region behaves according to a Markov chain. Let's assume that the probability of tomorrow being a wet day, given that today is wet, is 0.4. Similarly, the probability of tomorrow being a dry day, given that today is dry, is 0.7. If today is wet, the probability of tomorrow being a dry day is 0.6, and if today is dry, the probability of tomorrow being a wet day is 0.3.
(a) Find the transition matrix for this Markov chain. (2 marks)
(b) If Sunday is a dry day, what is the probability that Wednesday will be a rainy day? (2 marks)
(c) Find the steady-state vector and explain why it is an eigenvector for eigenvalue -1. (2 marks)
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