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AMarkov chain is a mathematical model that represents a system that transitions from one state to another over discrete time steps. It is a type of stochastic process with the Markov property, meaning that the future state of the system is independent of its past states given its present state. In other words, the Markov chain satisfies the Markov assumption. 

A Markov chain is characterized by a set of states and a transition probability matrix. The transition probability matrix describes the probabilities of moving from one state to another in a single time step. Each element P_ij of the matrix represents the probability of transitioning from state i to state j. Let X_t denote the state of the system at time t, where t takes on discrete values. The Markov property can be expressed as: 

         P(X_t+1 = j∣X_t =i, X_t−1 = i_t−1, …, X₀ = i₀) = P(X_t+1 = j∣X_t = i) 

In simpler terms, the probability of moving to the next state only depends on the current state and not on the entire history of states. Markov chains are used in various fields, including physics, biology, economics, and computer science, to model and analyze systems that exhibit probabilistic transitions between states. They are also a fundamental concept in the study of random processes and have applications in areas such as simulation, queuing theory, and statistical mechanics. 

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