Bayes Theorem Calculator

Calculate posterior probabilities using Bayes' theorem — perfect for medical testing, spam filters, and conditional probability problems.

Posterior Probability

P(A|B) = P(B|A) × P(A) / P(B)

Quick Presets

P(A) — Prior Probability

Base rate / prevalence of event A

P(B|A) — Likelihood / Sensitivity

Probability of B given A is true

P(B|¬A) — False Positive Rate

Probability of B given A is false

Posterior Probability P(A|B)

Prior P(A)

Evidence P(B)

Adjust the sliders or select a preset, then tap Calculate.

Prior vs Posterior Comparison

Step-by-step Solution

Bayes' Theorem

P(A|B) = P(B|A) × P(A) / P(B)

P(B) = P(B|A)×P(A) + P(B|¬A)×P(¬A)

P(A): Prior probability
P(B|A): Likelihood
P(A|B): Posterior probability

What Is Bayes' Theorem?

Bayes' theorem describes how to update the probability of a hypothesis based on new evidence. It's essential for medical diagnosis, spam filtering, and machine learning.

Common Use Cases

Medical Testing: disease prevalence + test accuracy

Spam Filtering: word frequency in spam vs. legitimate email

Quality Control: defect rate + inspection accuracy

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