A student has the habit that if she doesn't study one night,...

A student has the habit that if she doesn't study one night, she is $70 \%$ certain of studying the next night. Furthermore, the probability that she studies two nights in a row is $50 \%$. How often does she study in the long run?

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Key Concepts

Markov Chain

A Markov chain is a mathematical model that describes a process where the probability of transitioning to a future state depends only on the present state and not on the sequence of events that preceded it. This concept is used to model systems with memoryless properties in various fields such as statistics, economics, and physics.

Transition Probabilities

Transition probabilities are the key parameters in a Markov chain that describe the chance of moving from one state to another in a single time step. They form the elements of the state transition matrix, which provides the framework for analyzing the behavior of the system over time.

Stationary Distribution

The stationary distribution is a probability distribution over the states that remains unchanged as the system evolves over time. In a Markov chain, it represents the long-run fraction of time the process spends in each state. Finding this distribution requires solving a set of equilibrium equations based on the transition probabilities.

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