Larger menu. Consider a model of meal choice with four options: sushi, tacos, noodles and salad...

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Larger menu. Consider a model of meal choice with four options: sushi, tacos, noodles and salad (denote them s, t, n and a). Suppose that each day, with probability 1/2 you choose the same option as yesterday, otherwise you switch to one of the other 3 options chosen uniformly at random (i.e., each with the same probability). a) write down the transition probabilities of this model. Note that for each x, you should have YES p(x, y) = 1. b) assuming an arbitrary initial choice on day 1, find the probability that in the first four days, (i) you eat every type of meal. (ii) you eat exactly three different types of meals. (iii) on days 2,3 and 4 you eat a different meal than the one you had on day 1. Larger menu. Consider a model of meal choice with four options: sushi, tacos, noodles and salad (denote them s, t, n and a). Suppose that each day, with probability 1/2 you choose the same option as yesterday, otherwise you switch to one of the other 3 options chosen uniformly at random (i.e., each with the same probability). a) write down the transition probabilities of this model. Note that for each x, you should have YES p(x, y) = 1. b) assuming an arbitrary initial choice on day 1, find the probability that in the first four days, (i) you eat every type of meal. (ii) you eat exactly three different types of meals. (iii) on days 2,3 and 4 you eat a different meal than the one you had on day 1.