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Each entry in our matrix is non-negative, and each row straight across sums to one. If x is our initial row vector, then xP gives the probabilities for the particle's position after a minute and after k minutes, the vector xpk gives it, so the ith component of xPk gives the probability that the particle is at Ej after an elapse of k minutes. We reveal the Markov chain using an n x n transition matrix P, and a 1 x n row vector x. Positions Ei are the states of the chain. Now how does the particle move from one state to another? It gets from E4 to E1 in three minutes and goes from E4 to E6 in two minutes, but it cannot go from E1 to E4, because once the particle touches either the midgut or thorax, it stays there.

So our problem is not one of all actual probabilities but one when these probabilities are not zero.

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