Whenever Paul goes to his local supermarket, he visits the coffee shop for refreshments. He always chooses a blueberry muffin (B), a piece of carrot cake (C), a doughnut (D) or a scone (S). He never has the same item twice in a row. If he has a blueberry muffin on one visit, then on his next visit he has a doughnut. After carrot cake, he chooses either a doughnut or a scone on his next visit, with equal probability. After a doughnut, he is twice as likely to choose a scone on his next visit as carrot cake, and he is equally likely to choose carrot cake or a blueberry muffin. After a scone, he chooses either a doughnut or carrot cake with equal probability. Question 1: Write down the matrix of transition probabilities for Paul's choice of refreshments described above. Question 2: On one visit to the coffee shop, Paul has a doughnut. State probability that he has a carrot cake on the next visit. An what's the probability its in two visits time. An what is the probability that he has a carrot cake on the next visit and a doughnut in two visits time. Question 3: In the long run, on what proportion of visits to the coffee shop does Paul choose each of the four items (blueberry muffin, carrot cake, doughnut and scone)? Show all the steps in your working, and give your answers either as fractions or as decimals correct to three decimal places. And if Paul choses carrot cake on one visit, what is the expected number of visits till he chooses it again.

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