Bayes' Rule: Prior → Posterior
Probability · Class XII
Pick the prior P(H), sensitivity P(E|H) and false-positive rate P(E|¬H). Read the posterior.
- P(E)0.0594
- Posterior P(H|E)0.1667
- Likelihood ratio19.8
- Posterior vs prior
Formulas in this lab
- P(E)
- Posterior P(H|E)
- Likelihood ratio
Frequently asked questions
▶What is Bayes' rule?
Bayes' rule updates a prior probability $P(H)$ to a posterior $P(H|E)$ given evidence $E$: $P(H|E) = \dfrac{P(E|H)P(H)}{P(E)}$, where $P(E) = P(E|H)P(H) + P(E|\neg H)P(\neg H)$. It is the cornerstone of probabilistic reasoning.
▶How do I use the Bayes' rule lab?
Slide the prior $P(H)$, sensitivity $P(E|H)$ and false-positive rate $P(E|\neg H)$. The lab returns the total probability $P(E)$ and the posterior $P(H|E)$. Try prior $0.01$, sensitivity $0.99$, false-positive $0.05$: posterior $\approx 0.17$ — surprisingly low!
▶Why does a rare disease give a low posterior even with a $99\%$ accurate test?
Because the prior $P(H) = 0.01$ is tiny: most positive results come from the much larger non-diseased population. Even with sensitivity $0.99$ and specificity $0.95$, only about $1$ in $6$ positives are true. This is the base-rate fallacy — a JEE Probability favourite.
▶Where is Bayes' rule used in real applications?
Spam filtering, medical diagnostics, weather forecasting (assimilating new data), and Aadhaar's fingerprint matching all use Bayes. Modern AI like ChatGPT trains using Bayesian-style updates of model parameters. The rule is everywhere data meets uncertainty.