Suppose that a contagious virus spreads among three individuals that are all neighbors with each other as follows. Whether or not an individual is sick on day n + 1 depends only on how many sick neighbors she had on day n. At the end of day n, if an individual has zero, one, or two sick neighbors, then she is sick at the beginning of day n +1 with probability Po, P1, or P2, respectively. The condition (sick/well) of an individual does not change throughout a day. Suppose that X = (X, n >=0) is a Markov chain for which Xn, denotes the number of sick individuals during day n. Required:
Find the transition matrix of X.