the Markov property
/ MAR-kov /
Imagine a frog hopping between lily pads. To predict where it jumps next, do you need its entire travel diary — every pad it ever sat on — or just the pad it is on right now? The Markov property is the assumption that the present pad is all that matters: once you know where the frog is now, the whole past adds nothing to your prediction of the future. The process has, in this precise sense, no memory.
Stated carefully, a process X_0, X_1, X_2, ... has the Markov property if P(X_(n+1) = j given X_n = i AND X_(n-1), ..., X_0) = P(X_(n+1) = j given X_n = i). In words: conditioning on the entire history gives exactly the same answer as conditioning on the most recent state alone. The current state acts as a complete summary, a single sufficient stand-in for everything that came before. Past and future are independent once the present is held fixed.
This is a modelling assumption, not a law of nature, and it is what makes huge stochastic systems tractable: instead of tracking an ever-growing history you carry only the current state. Many real systems are not exactly Markov, but a clever choice of state often makes them so — if yesterday's value also matters, fold it into the state by letting the state be the pair (today, yesterday). The trick of probability modelling is choosing a state rich enough that the future really does depend only on the present.
A board game where you move by a die roll is Markov in your square: your next square depends only on where you stand now plus the roll, never on the path you took to get there. But a card game where played cards are gone is NOT Markov in 'the card on the table' alone — the deck's history matters — unless you enlarge the state to include which cards remain.
Whether a process is Markov depends on how you choose its state; enrich the state and memory can vanish.
Memoryless does not mean the future is independent of the past — only that the past adds nothing ONCE the present is known. Past and future are conditionally independent given the present, not unconditionally independent.