1. Enter the prior probability
The prior represents how likely A is before the new evidence B is observed.
Conditional probability calculator
Bayes’ Theorem Calculator
Update the probability of an event after observing evidence. Use direct Bayes mode when the evidence probability is known, or binary-test mode when sensitivity and false-positive rate are available.
Percent and decimal inputs
Bayes factor and odds
Expected-count interpretation
Posterior probability
32.14%
Prior
5.00%
Bayes factor
9.00×
Calculate posterior probability
Choose a calculation mode, enter the probabilities, and review how the evidence changes the prior belief.
Bayes’ theorem updates a prior probability using evidence. The
result is only as reliable as the assumptions, data quality,
likelihood estimates, and independence conditions used in the model.
How the Bayes’ Theorem Calculator works
Bayes’ theorem combines a prior probability with the likelihood of observed evidence. In binary mode, the calculator also derives the total evidence rate by considering evidence among both A and not-A cases.
2. Describe the evidence
Enter P(B|A), then either provide P(B) directly or enter the false-positive rate P(B|not A).
3. Review the posterior
Compare the updated probability with prior odds, posterior odds, Bayes factor, probability lift, and expected counts.
Bayes’ Theorem
P(A|B) = P(B|A) × P(A) ÷ P(B)
Total Evidence Probability
P(B) = P(B|A)P(A) + P(B|not A)P(not A)
Posterior Odds
Posterior Odds = Prior Odds × Bayes Factor
Metric
Meaning
How to interpret it
Posterior probability
Updated belief after evidence
Compare it with the prior probability to see whether the evidence raises or lowers the probability of A.
Bayes factor
Evidence strength in odds form
Values above 1 support A, values below 1 support not-A, and values near 1 provide little discrimination.
False-discovery probability
Probability A is false despite B
This is the complement of the posterior probability and is especially useful when base rates are low.
Expected counts
Population-based interpretation
Counts make base-rate effects easier to see by showing true and false evidence cases in an illustrative population.
Frequently asked questions
Important details for calculating and interpreting Bayes’ theorem.
What is Bayes’ theorem?
Bayes’ theorem is a probability rule used to update the probability of an event after observing related evidence.
What is a prior probability?
A prior probability is the estimated probability of A before considering the new evidence B.
What is a posterior probability?
The posterior probability is the updated probability of A after taking the observed evidence B into account.
What does P(B|A) mean?
P(B|A) is the probability of observing evidence B when A is true. In diagnostic testing, it is often called sensitivity.
Why is the false-positive rate important?
It measures how often B occurs when A is false. A high false-positive rate can greatly reduce the posterior probability, especially when A is rare.
What is a Bayes factor?
In binary mode, the Bayes factor is P(B|A) divided by P(B|not A). It shows how much more likely the evidence is under A than under not-A.
Can evidence reduce probability?
Yes. If the evidence is more likely when A is false than when A is true, the posterior probability can be lower than the prior.
Where is Bayes’ theorem used?
It is used in medical testing, spam detection, machine learning, fraud detection, forecasting, legal reasoning, reliability analysis, and scientific inference.