A Markov chain is a stochastic model that represents a sequence of events in which the probability of each event depends only on the state attained in the previous event. Key characteristics of a Markov chain include the Markov property, which states that future states depend only on the current state and not on the past states, and the finite or countably infinite set of possible states. Consider a real-world scenario like a person's daily commute to work, which involves multiple modes of transportation. Let's simplify this scenario to include three states: walking, taking the bus, and riding a bike. 1. Walking: The person starts at home and walks to the bus stop. 2. Bus: At the bus stop, the person boards the bus. 3. Biking: After getting off the bus, the person unlocks their bike from the bike rack and rides the rest of the way to work. Each transition between states can be represented as a probabilistic event. For example, the probability of transitioning from walking to taking the bus might depend on factors like the time of day (e.g., rush hour), weather conditions, or personal preferences. Similarly, the probability of transitioning from the bus to biking could