What is Bayes theorem?

Bayes theorem states P(A|B) = P(B|A) × P(A) / P(B). It provides a mathematical framework for updating the probability of a hypothesis (A) after observing evidence (B). The prior P(A) represents your belief before the evidence; the posterior P(A|B) represents your updated belief after it.

What is the difference between prior and posterior probability?

The prior P(A) is your initial belief about hypothesis A before any evidence is considered. The posterior P(A|B) is the updated belief after incorporating evidence B. Bayes theorem bridges these two by weighting the prior with the likelihood of the evidence.

What is P(B) and how do I calculate it?

P(B) is the total (marginal) probability of observing the evidence regardless of whether A is true. It can be computed using the law of total probability: P(B) = P(B|A)×P(A) + P(B|A')×P(A'), where A' is the complement of A. If you only know P(B|A) and the priors, use that formula first.

Why must P(B|A) × P(A) not exceed P(B)?

The joint probability P(A∩B) = P(B|A)×P(A) represents both A and B occurring simultaneously, which is a subset of B alone. Therefore P(A∩B) ≤ P(B) always. If your inputs violate this, at least one value is inconsistent.

Can the posterior probability be higher than the prior?

Yes. If the evidence is more likely under hypothesis A than in general (P(B|A) > P(B)), the posterior P(A|B) will exceed the prior P(A). Conversely, evidence that is unlikely given A will lower the posterior below the prior.

What is the base rate fallacy?

The base rate fallacy occurs when people ignore the prior P(A) and focus only on the likelihood P(B|A). For example, a 99% accurate test for a rare disease (P(A)=0.001) still yields a low posterior because most positive results are false positives. Bayes theorem quantifies this correctly.

Bayes Theorem Calculator

Name: Bayes Theorem Calculator
Availability: InStock
Author: Nham Vu

Calculate posterior probability P(A|B) using Bayes theorem from prior P(A), likelihood P(B|A), and marginal probability P(B).

Bayes Theorem Formula

P(A|B) = P(B|A) × P(A) ÷ P(B)

P(A|B) — posterior

P(B|A) — likelihood

P(A) — prior

P(B) — marginal

Enter Probabilities (0 to 1)

P(A) — Prior probability of hypothesis A

Probability that A is true before observing evidence B.

P(B|A) — Likelihood of evidence B given A

How likely is the evidence if hypothesis A is true?

P(B) — Marginal probability of evidence B

Overall probability of observing B regardless of A. Use P(B) = P(B|A)×P(A) + P(B|A')×P(A') if unknown.

Quick Examples

Enter probabilities and click Calculate to see results.

Summary

Calculate posterior probability P(A|B) using Bayes theorem from prior P(A), likelihood P(B|A), and marginal probability P(B).

How it works

Enter P(A), the prior probability of your hypothesis before seeing evidence.
Enter P(B|A), the likelihood — how probable the evidence is if the hypothesis is true.
Enter P(B), the total marginal probability of observing the evidence.
Click "Calculate" and the tool applies the formula P(A|B) = P(B|A) × P(A) / P(B).
Review the posterior probability, complement, joint probability, and the plain-English interpretation.
Adjust inputs to explore how the posterior changes as you update your beliefs.

Use cases

Medical diagnosis: find P(disease | positive test) from test sensitivity and disease prevalence.
Spam filtering: compute P(spam | keyword) from training data likelihoods.
Quality control: estimate P(defective | failed inspection) from historical defect rates.
Finance: update P(market crash | indicator) as new economic signals arrive.
Legal reasoning: evaluate how new evidence shifts the probability of guilt.
Machine learning: understand naive Bayes classifiers and probabilistic model updates.
Drug testing: assess the true positive rate given test accuracy and prevalence.
A/B testing: convert observed conversion rates into posterior belief about variant performance.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu