Bayes’ theorem is a bookkeeping rule for reversing conditional probabilities. If you know how likely the evidence is under a hypothesis, and how common the hypothesis is beforehand, you can update after seeing the evidence. The algebra is short; the discipline is in stating the events clearly.
The formula
$$P(H\mid E)=\frac{P(E\mid H)P(H)}{P(E)}$$ Here $H$ is a hypothesis and $E$ is evidence. The denominator $P(E)$ expands by total probability when $H$ and its complement partition the space: $P(E)=P(E\mid H)P(H)+P(E\mid H^c)P(H^c)$.
Think in counts
Example Imagine 10,000 people; 1% have a condition ($H$). A test is 99% sensitive ($P(E\mid H)=0.99$) and 95% specific, so false positive rate $P(E\mid H^c)=0.05$. About 100 people have $H$; roughly 99 test positive. Among the 9,900 without $H$, about 495 false positives. So among ~594 positive tests, only ~99 truly have $H$ — a bit under 17%, not 99%. The test is informative, but the rare base rate dominates the headline sensitivity.
What “prior” and “posterior” mean
$P(H)$ is the prior — your probability before seeing $E$. $P(H\mid E)$ is the posterior — after updating. Likelihood $P(E\mid H)$ is not a probability distribution over hypotheses by itself; it measures how well each hypothesis predicts the data.
For the surrounding map, see Probability & statistics.
Citations & further reading
- OpenIntro Statistics (open textbook) — conditional probability and Bayes. openintro.org
- Khan Academy, Bayes’ theorem lessons. khanacademy.org
- Stanford CS109 probability notes (Bayes applications). web.stanford.edu/class/cs109
- Thinking in counts pedagogy popularized in medical decision teaching; see also Gigerenzer’s work on natural frequencies (overview articles via scholarly search).