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We're on Markov and we're considering the Markov chain on $\{A,B,C,D,E\}$ with a transition matrix $P$. I am asked to find $P^3(D,D)$ but I am unfamiliar with what the notation stands for.

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It probably means the probability that the Markov chain ends at $D$, given that you started at $D$ and took three steps. So, you can find this probability by cubing the transition matrix and then looking at the value in row $D$ and column $D$.

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