Bayes' theorem
Updated: 5/24/2026, 7:31:31 PM Wikipedia source
Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes (), gives a mathematical rule for inverting conditional probabilities, allowing the probability of a cause to be found given its effect. For example, with Bayes' theorem, the probability that a patient has a disease given that they tested positive for that disease can be found using the probability that the test yields a positive result when the disease is present. The theorem was developed in the 18th century by Bayes and independently by Pierre-Simon Laplace. One of Bayes' theorem's many applications is Bayesian inference, an approach to statistical inference, where it is used to invert the probability of observations given a model configuration (i ., the likelihood function) to obtain the probability of the model configuration given the observations (i ., the posterior probability).
Tables
| SymptomCancer | Yes | No | Total | |
| Yes | 1 | 0 | 1 | |
| No | 10 | 99989 | 99999 | |
| Total | 11 | 99989 | 100000 |
| Background Proposition | B | ¬ (not B) | Total |
| A | P ( B | A ) ⋅ P ( A ) {\displaystyle P(B|A)\cdot P(A)} = P ( A | B ) ⋅ P ( B ) {\d | P ( ¬ B | A ) ⋅ P ( A ) {\displaystyle P( eg B|A)\cdot P(A)} = P ( A | ¬ B ) ⋅ P ( ¬ | P ( A ) {\displaystyle P(A)} |
| ¬ (not A) | P ( B | ¬ A ) ⋅ P ( ¬ A ) {\displaystyle P(B| eg A)\cdot P( eg A)} = P ( ¬ A | B ) ⋅ P ( | P ( ¬ B | ¬ A ) ⋅ P ( ¬ A ) {\displaystyle P( eg B| eg A)\cdot P( eg A)} = P ( ¬ A | ¬ B ) | P ( ¬ A ) {\displaystyle P( eg A)} = 1 − P ( A ) {\displaystyle 1-P(A)} |
| Total | P ( B ) {\displaystyle P(B)} | P ( ¬ B ) = 1 − P ( B ) {\displaystyle P( eg B)=1-P(B)} | 1 |
References
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