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Bayes’ Theorem Calculator

Update probability with evidence. Enter a prior, a likelihood, and a false-positive rate—then get the posterior P(A|B), the evidence probability P(B), and a clear breakdown in decimal and percent.

Posterior P(A|B) Evidence P(B) Base Rate A History + CSV

Bayesian Probability Evaluator

Compute P(A|B) using Bayes’ theorem with a simple two-case model: A vs ¬A. Save results to history and export to CSV.

How common A is before seeing evidence B (base rate).
How often evidence appears when A is true (e.g., sensitivity).
How often evidence appears when A is false (e.g., 1 − specificity).

Quick Steps

  1. Choose whether you want to enter values as percents or decimals.
  2. Enter P(A), P(B|A), and P(B|¬A) using the same format.
  3. Press Calculate to get P(A|B), P(B), and P(¬A|B).
  4. Use History to compare scenarios and export results to CSV.
Tip: A common “positive test” setup is A = condition present and B = test positive. Then P(A|B) is the probability you truly have the condition given a positive result.
Concept Symbol Plain Meaning Example Interpretation
Prior P(A) How likely A is before evidence Prevalence of a condition
Likelihood P(B|A) How often B happens if A is true Sensitivity: positive if condition present
False Positive P(B|¬A) How often B happens if A is false 1 − specificity: positive if condition absent
Evidence P(B) Overall probability of observing B Chance of a positive test in the population
Posterior P(A|B) Updated probability of A after seeing B Chance condition is present given positive test
Complement ¬A “Not A” No condition present

Formula Breakdown

  1. Compute the complement: P(¬A) = 1 − P(A).
  2. Compute evidence: P(B) = P(B|A)P(A) + P(B|¬A)P(¬A).
  3. Compute posterior: P(A|B) = P(B|A)P(A) ÷ P(B).
  4. Compute the alternative: P(¬A|B) = 1 − P(A|B).
Valid ranges: probabilities must be between 0 and 1 (decimal) or 0 and 100 (percent). If P(B) becomes 0, the posterior is undefined and the calculator will show an error.
Your Bayes calculations will appear here after you run the calculator.

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