Bayes' Formula
For a collection of disjoint events covering with for all , and with ,
Proposition 2.14 (Bayes' Formula)
Consider to be a collection of disjoint events in such that and for all . Then for any where ,
Proof
By definition of conditional probability, . The numerator rewrites as , and the law of total probability expands the denominator as .
False positives for a rare condition
A rare condition affects of the population: . A test has true positive rate and false positive rate . Testing positive,
Because the condition is rare, the posterior probability stays small even though the test is highly reliable.
Related
Stated in
- Proposition 2.14 (Bayes' Formula)§2.4 Conditional Probability