Transition probabilities

Transition probabilities describe the likelihood of moving from one state to another in a system that evolves over time. They are the numerical heart of Markov chains and similar stochastic models, where the future depends only on the present state, not on history.

In practice, these probabilities appear as a matrix: each row represents a current state, each column a possible next state, and each entry gives the chance of that transition occurring. They answer questions like "if it rains today, what's the probability of rain tomorrow?" or "if a customer buys product A, how likely are they to buy product B next?"

Transition probabilities appear across physics, molecular biology, ecosystem dynamics, chemical reaction rates, and anomaly detection. They're foundational to temporal reasoning about systems that change—whether you're modeling stellar evolution, predicting path-dependent outcomes, or training probabilistic models in machine learning.

Computing them often requires manual calculation or fitting to observed data. The elegance lies in their simplicity: from a small set of transition probabilities, you can compute the entire future behavior of a system through composition and recursion.

Related

Markov Chain, Stochastic Process, Matrix, Probability Distribution, State Space