How does the Bayes Theorem Calculator work?

P(A|B) = P(B|A) × P(A) / P(B) Then: P(A) = prior probability (before evidence) Then: P(B|A) = likelihood (probability of evidence given hypothesis) Then: P(A|B) = posterior probability (after evidence).

Can you give an example of how to use the Bayes Theorem Calculator?

Example: Input Disease prevalence 1%, test sensitivity 99%, specificity 95% gives a result of P(disease | positive test) ≈ 16.7% (Despite 99% accuracy, only 1 in 6 positives are true positives).

Is the Bayes Theorem Calculator free to use?

Yes — completely free with no registration, download, or subscription required. All calculations happen instantly in your browser.

How accurate is the Bayes Theorem Calculator?

Our Bayes Theorem Calculator uses verified mathematical formulas and is accurate to multiple decimal places. Results are calculated in real-time using the same methods used by professionals.

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Bayes Theorem Calculator

P(A|B)Bayes Theorem Calculator

Classic medical screening example: what is the probability of disease given a positive test?

Prior P(D) %

Population prevalence

Sensitivity %

P(+|Disease)

Specificity %

P(−|No disease)

Bayes' Theorem describes how to update the probability of a hypothesis given new evidence. Named after Rev. Thomas Bayes (1702–1761), it is the mathematical foundation of Bayesian statistics and is central to medical diagnosis, spam filtering, machine learning, and forensic science. The theorem shows why even a highly accurate test can produce mostly false positives when the disease is rare — this is the base rate fallacy.

Symbol	Name	Unit	Description
P	P(B\|A)×P(A) + P(B\|¬A)×P(¬A) — the total probability	—	P(B\|A)×P(A) + P(B\|¬A)×P(¬A) — the total probability
B	P(B\|A)×P(A) + P(B\|¬A)×P(¬A) — the total probability	—	P(B\|A)×P(A) + P(B\|¬A)×P(¬A) — the total probability
A	posterior probability (after evidence)	—	posterior probability (after evidence)

Term	Symbol	Meaning
Prior	P(D)	Probability of disease before testing
Sensitivity	P(+\|D)	True positive rate
Specificity	P(−\|¬D)	True negative rate
False positive rate	P(+\|¬D)	1 − specificity
Posterior	P(D\|+)	Probability of disease given positive test
Positive predictive value	PPV = P(D\|+)	Clinical relevance of a positive result

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