Probability theory

Probability theory is the mathematical study of random events and uncertainty. It formalizes probability through models, rules, and structures used to describe and analyze uncertain outcomes.

Probability space: A mathematical model consisting of outcomes, events, and a probability measure. It provides the formal setting in which probability theory assigns likelihoods.
Bayes' theorem: A rule relating conditional probabilities in opposite directions. It updates a hypothesis’s probability using observed evidence.
Statistical inference: The use of data and statistical models to draw conclusions about populations or processes. It uses probability to quantify uncertainty in conclusions drawn from samples.
Problem of points: A historical problem asking how stakes should be divided when a game of chance ends early. Its solution helped prompt the mathematical study of gambling probabilities.
Frequentist interpretation of probability: An interpretation identifying probability with limiting relative frequency in repeated trials. It contrasts with interpretations that treat probability as degrees of belief.
Sample space: The set of all possible outcomes of a random experiment. Events and probability measures are defined relative to this set.
Law of total probability: A formula expressing an event’s probability as a sum over a partition of possible cases. It combines conditional probabilities across mutually exclusive cases.
Bayesian statistics: An approach to statistical inference that updates probability distributions using evidence and Bayes’ theorem. It treats unknown quantities probabilistically and revises beliefs as data arrive.
Blaise Pascal: A French mathematician, physicist, and philosopher who helped establish early probability theory. His correspondence with Fermat addressed foundational problems of chance.
Bayesian interpretation of probability: An interpretation treating probability as a coherent degree of belief about uncertain propositions. It allows probabilities for hypotheses, not only repeatable experimental outcomes.

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