Conditional probability

Conditional probability measures the chance of an event when another event is known to have occurred. For a condition with positive probability, it is the probability of their intersection divided by the probability of the condition.

Conditional probability formula: The identity P(A|B) = P(A ∩ B) / P(B), valid when P(B) is positive. It expresses restriction to condition B as renormalizing the probability of A ∩ B.
Probability space: A mathematical model consisting of outcomes, a collection of events, and a probability measure. Conditional probability modifies probabilities within this underlying model.
Bayesian inference: A statistical approach that updates probability distributions using observed data and Bayes' theorem. Observed evidence becomes a condition for revised probabilities about hypotheses.
Joint probability distribution: A probability distribution describing the combined outcomes of two or more random variables. A joint distribution describes variables together; conditioning extracts a distribution at specified information.
Regular conditional probability: A probability kernel describing conditional distributions, including conditioning on random variables with zero-probability values. It provides a rigorous framework where conditioning on a point defeats the elementary ratio formula.
Conditional distribution: A probability distribution describing a random variable when another variable or event is specified. It extends event conditioning to distributions of random variables.
Sample space: The set of possible outcomes represented by a probability model. Conditioning can be pictured as narrowing attention to outcomes compatible with the condition.
Diagnostic testing: The evaluation of tests that classify individuals or samples as positive or negative for a condition. The probability of disease given a positive result depends on conditional probabilities and prevalence.
Marginal probability distribution: A distribution for one variable obtained from a joint distribution by summing or integrating over others. Marginalization removes variables, whereas conditioning fixes information about them.
Disintegration theorem: A measure-theoretic result that decomposes a joint distribution into a marginal distribution and conditional distributions. It underpins the existence and structure of conditional distributions in general spaces.

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