Bayes' Theorem

Bayes' theorem lets you reverse the direction of a conditional probability: $$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$ Or in the full "partition" form with hypotheses $H_1, \ldots, H_n$: $$P(H_i|E) = \frac{P(E|H_i) \cdot P(H_i)}{\sum_{j=1}^{n} P(E|H_j) \cdot P(H_j)}$$ Intuition: You observe evidence $E$. You want to know which hypothesis caused it. Bayes says: start with your prior belief $P(H_i)$, multiply by how likely the evidence is under that hypothesis $P(E|H_i)$, then normalize so everything sums to 1. Evidence that is more likely under hypothesis $H_i$ pushes you toward believing $H_i$. Concrete example: A medical test is 99% sensitive (catches 99% of sick people) and 95% specific (5% false positive rate). The disease prevalence is 1%. You test positive. What's the probability you're actually sick? $$P(\text{sick}|+) = \frac{0.99 \times 0.01}{0.99 \times 0.01 + 0.05 \times 0.99} = \frac{0.0099}{0.0099 + 0.0495} \approx 16.7\%$$ Despite a "99% accurate" test, the low base rate means most positives are false alarms! $\frac{10}{60}$ ≈ 17% When to use: Any problem that gives you $P(B|A)$ but asks for $P(A|B)$. Medical diagnostics, spam filtering, updating beliefs with new evidence, and nearly all Bayesian statistics. This is a fundamental technique — arguably the most fundamental result in probability. It underpins Bayesian inference, machine learning, and decision-making under uncertainty.