Understanding the Core Concept
A Markov chain is a mathematical model for a sequence of possible events in which the probability of each event depends only on the state achieved in the previous event. This property is known as the "memoryless" property, or the Markov property, meaning that the past history beyond the immediate previous state does not influence the future.
Key Components of a Markov Chain
A Markov chain is defined by its states (the possible outcomes or conditions) and its transition probabilities. These probabilities dictate the likelihood of moving from one state to another. The collection of these probabilities forms a transition matrix, which is central to analyzing the behavior of the chain.
A Practical Example: Weather Prediction
Consider a simplified weather model with two states: "Sunny" and "Rainy." If today is Sunny, there might be a 90% chance it's Sunny tomorrow and a 10% chance it's Rainy. If today is Rainy, there might be a 50% chance it's Sunny tomorrow and a 50% chance it's Rainy. This sequence of weather, where tomorrow's weather only depends on today's, forms a Markov chain.
Importance and Applications
Markov chains are crucial across various fields, including physics, chemistry, biology, economics, and computer science. They are used for modeling queueing systems, analyzing gene sequences, predicting stock prices, ranking web pages (like Google's PageRank algorithm), and understanding physical processes where future states depend only on the present.