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Conditional Probability

Defines Conditional Probability, conditional probabilities

The conditional probability of event A given event B, written P(A | B), is the probability of A occurring given that B has occurred. It is defined as P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. Conditioning restricts the sample space to outcomes in B and rescales probabilities accordingly.

Conditional probability leads to Bayes’ theorem: P(A | B) = P(B | A) · P(A) / P(B). This relates the probability of a hypothesis given evidence to the probability of evidence given the hypothesis, and is the foundation of Bayesian inference. The prior P(A) is updated by the likelihood P(B | A) to produce the posterior P(A | B).

Two events are independent when conditioning changes nothing: P(A | B) = P(A), which is equivalent to P(A ∩ B) = P(A) · P(B). Conditional probability for more than two events gives rise to chains: P(A ∩ B ∩ C) = P(A) · P(B | A) · P(C | A ∩ B), a decomposition used in probabilistic graphical models and sequential reasoning.

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@misc{emsenn2026-conditional-probability,
  author    = {emsenn},
  title     = {Conditional Probability},
  year      = {2026},
  url       = {https://emsenn.net/library/math/domains/probability/terms/conditional-probability/},
  publisher = {emsenn.net},
  license   = {CC BY-SA 4.0}
}