Urn Models

Urn models represent sampling through explicit initial contents and replacement or reinforcement rules. Those rules determine dependence, conditional probabilities, and long-run behavior.

Urn models make sampling rules visible through a population of colored balls. Drawing with replacement from a fixed composition can represent independent sampling, while removing balls or reinforcing the selected color changes subsequent probabilities. The update rule is part of the probability model, not a detail to add afterward.

Exchangeability means that the probability of a sequence is unchanged by rearranging its order; it does not mean independence. The classic Pólya urn is exchangeable despite dependent draws. Reinforcement changes predictive probabilities, yet its standard balanced form does not imply that the first color drawn inevitably takes over the entire urn.

When to use it

When understanding whether early outcomes create path dependence; when initial conditions may disproportionately determine final outcomes; when the difference between independent and dependent sampling matters for strategy; when formalizing intuitions about how composition affects probability.

How it can help

Define the contents and update mechanism, then calculate how each observation changes subsequent probabilities. Distinguish independence from exchangeability.