Definition
A Markov chain is a stochastic process that moves between a discrete set of states, where the probability of transitioning to the next state depends only on the current state — not on how the process arrived there. This 'memoryless' property is the Markov property, named after Andrey Markov who introduced it around 1906.
A Markov chain is fully specified by its set of states and a transition matrix giving the probability of moving from each state to each other state.
Why it matters
How it works
A Markov chain is defined by states S1, S2, ..., Sn and a transition matrix P where P[i][j] = probability of moving from Si to Sj in one step. Starting from some initial state, the chain evolves stochastically, with all future probabilities determined by the matrix and the current state.
For example, a simple two-state weather chain (Sunny, Rainy) might have transition probabilities P(Sunny -> Sunny) = 0.8, P(Sunny -> Rainy) = 0.2, P(Rainy -> Sunny) = 0.4, P(Rainy -> Rainy) = 0.6. From this matrix you can compute the probability of any sequence of weather, the long-run fraction of sunny days, and how quickly forgetting of the initial state occurs.