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Formal Sciences
Probability
1763
Intermediate

Bayes' Theorem

P(A∣B)=P(B∣A) P(A)P(B)P(A|B) = \frac{P(B|A)\, P(A)}{P(B)}

Updates your belief about A after observing evidence B—rational reasoning under uncertainty.

By Thomas Bayes, Pierre-Simon Laplace

Formal Sciences
Bayes' Theorem
1763 · Thomas Bayes
Classroom Ready
89%

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Story PortalWho discovered this?Discover how Thomas Bayes and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — spam filters and medical diagnostics.Math PortalWhat does it derive from?Trace the mathematical lineage from conditional probability.Future PortalWhere is it going?Bayesian deep learning
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Why it matters: Foundation of modern statistics, medical diagnosis, spam filters, and AI reasoning.

Discoverers: Thomas Bayes, Pierre-Simon Laplace (1763)

What does it mean?

Updates your belief about A after observing evidence B—rational reasoning under uncertainty.

Why should I care?

Foundation of modern statistics, medical diagnosis, spam filters, and AI reasoning.

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Variables & Units

SymbolNameUnitMeaning
P(A∣B)P(A|B)Posterior—Updated probability of A given B
P(B∣A)P(B|A)Likelihood—Probability of B given A
P(A)P(A)Prior—Initial belief about A

Worked Example

If 1% have disease and test is 99% accurate, P(disease|positive) ≈ 50% with 1% prevalence.

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Sources & further reading

Search WikipediaSearch Wolfram MathWorldOpen in Wolfram|AlphaWatch explainers on YouTube

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Equation Universe

Bayes' Theorem

P(A∣B)=P(B∣A) P(A)P(B)P(A|B) = \frac{P(B|A)\, P(A)}{P(B)}

Real-world impact

Life sciences

Mathematical models drive medicine and biotech.

Photo: Unsplash — laboratory

Updates your belief about A after observing evidence B—rational reasoning under uncertainty.

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