Bayes' theorem
Update a probability when new evidence arrives.
Library note. Check the assumptions and further reading before applying a formula.
What Is This?
Bayes' theorem connects two conditional probabilities. P(A) is the probability before observing B. P(B given A) describes how likely the evidence would be if A were true. Dividing by the overall chance of B gives the updated probability of A. The base rate matters: even a convincing-looking observation can be common enough among alternatives to give a weak conclusion.
Try an example
Out of 1,000 items, suppose 10 are faulty. A test flags 9 of those and also flags 99 good items. Among the 108 flagged items, only 9 are faulty: about 8.3%.
Where it needs care
The example is hypothetical. Real updates depend on reliable base rates and likelihoods, and the evidence must have nonzero probability under the model.
Historical Context
Thomas Bayes' work on inverse probability was published after his death in 1763. Pierre-Simon Laplace developed the approach more broadly.
Real-World Applications
- Interpret diagnostic test results.
- Update forecasts as observations arrive.
