In mathematics, a Markov chain, named after Andrey Markov, is a discrete random process with the Markov property. A discrete random process means a system which can be in various states, and which changes randomly in discrete steps. It can be helpful to think of the system as evolving through discrete steps in time, although strictly speaking the "step" may have nothing to do with time. The Markov property states that the probability distribution for the system at the next step (and in fact at all future steps) only depends on the current state of the system, and not additionally on the state of the system at previous steps. Since the system changes randomly, it is generally impossible to predict the exact state of the system in the future. However, the statistical properties of the system at a great many steps in the future can often be described. In many applications it is these statistical properties that are important.
The changes of state of the system are called transitions, and the probabilities associated with various state-changes are called transition probabilities.
An example of a Markov chain is a random walk on the number line where, at each step, the position may change by +1 or −1 with equal probability. From any position there are two possible transitions: to the next or previous integer. The transition probabilities depend only on the current position, not on the way the position was reached. For example, if the current position is 5, then the transition to 6 has a probability of 0.5, regardless of any prior positions.
Another example is the dietary habits of a creature who eats only grapes, cheese or lettuce, and whose dietary habits conform to the following (artificial) rules: it eats exactly once a day; if it ate cheese yesterday, it will not today, and it will eat lettuce or grapes with equal probability; if it ate grapes yesterday, it will eat grapes today with probability 1/10, cheese with probability 4/10 and lettuce with probability 5/10; finally, if it ate lettuce yesterday, it won't eat lettuce again today but will eat grapes with probability 4/10 or cheese with probability 6/10. This creature's eating habits can be modeled with a Markov chain since its choice depends on what it ate yesterday, not additionally on what it ate 2 or 3 (or 4, etc.) days ago. One statistical property one could calculate is the expected percentage of the time the creature will eat cheese over a long period.
A series of independent events—for example, a series of coin flips—does satisfy the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next step depends non-trivially on the current state.
Many other examples of Markov chains exist.