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Compute multi-step Markov probabilities and estimate transitions

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This question evaluates understanding of discrete-time Markov chains, computation of multi-step transition probabilities from a one-step transition matrix, and empirical estimation of transition matrices from observed state-pair data including handling states with zero outgoing transitions.

Overview: This question evaluates understanding of discrete-time Markov chains, computation of multi-step transition probabilities from a one-step transition matrix, and empirical estimation of transition matrices from observed state-pair data including handling states with zero outgoing transitions.