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4. (30 points) Amia goes grocery shopping at either Kroeger (K), Buschs (B), or Plum Market (P). If she goes to Kroeger today, next time she will either go to Kroeger again with probability 0.2, she will go to Buschs with probability 0.5, or otherwise she will go to Plum Market. If she goes to Buschs today, then she will go to Kroeger, Buschs, or Plum Market next time with probabilities 0.4, 0.2, and 0.4, respectively. If she goes to Plum Market today, then next time she will go to Kroeger with probability 0.6, Buschs with probability 0.3, or Plum Market with probability 0.1. (a) (10 points) Given that for her first fall semester groceries she either goes to Kroeger with probability 0.4, Buschs with probability 0.3, or Plum Market with probability 0.3, what is the probability that she will go to Kroeger on her 5th grocery store visit? (b) (5 points) On what portion of the visits will she not go to Kroeger? Make sure to list al limiting probabilities equations (c) (5 points) Assume that Amia goes grocery shopping twice a week. She spends $40 when she buys groceries from Kroeger, $50 when she buys groceries from Buschs and $60 when she buys groceries from Plum Market. On average, how much does she spend in groceries per month? Assume each month has 4 weeks (d) (10 points) If Amia buys from Buschs today, what is the probability that she buys form Plum Market in none of the next 3 visits? (Hint: This is the equivalent of asking that she goes to Kroeger or Buschs in all of the next 3 visits.)

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