Bayes' Theorem

Bayes' theorem relates conditional probabilities, letting you reverse the direction of conditioning:

$P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)}$

Given prior $P(A)$ (your belief before evidence) and likelihood $P(B \mid A)$, compute the posterior $P(A \mid B)$ — your updated belief after seeing $B$.

Classic medical-test example: disease prevalence 1%, test sensitivity 99%, false-positive rate 1%. The probability of disease given a positive test:

$\frac{0.99 \cdot 0.01}{0.99 \cdot 0.01 + 0.01 \cdot 0.99} = \frac{1}{2}$

Despite a 99% accurate test, a positive result means only 50% chance of disease — because the disease is rare. The "base rate fallacy" (forgetting the prior) is the most common Bayes mistake.

Bayes powers Bayesian inference, naïve Bayes classifiers, spam filters, and forensic reasoning.