Power Calculators

🧠 Bayes’ Theorem Calculator (Posterior Probability)

Why a positive test for a rare disease is often a false alarm — worked out with Bayes.

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Enter details

e.g. how common a condition is: 1% = 0.01
test sensitivity / true-positive rate
false-positive rate (1 − specificity)

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Fill in the details and click Calculate to see your results.

Formula at a glance

Bayes’ Theorem Calculator: Bayes’ theorem — P(A|B) = P(B|A)·P(A) / P(B). Example: Rare condition, accurate test → P(A | B) — posterior: 0.161017 (16.1017%)
How the Bayes’ Theorem Calculator works: Bayes’ theorem — P(A|B) = P(B|A)·P(A) / P(B)

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📘 About the Bayes’ Theorem Calculator

Bayes’ theorem updates a probability when new evidence arrives. It answers questions like: if a test is 95% accurate and 1% of people have the condition, what is the chance that someone who tests positive really has it? Surprisingly often, the answer is low — because false positives from the healthy majority outnumber true positives.

Enter the prior and the two likelihoods. The calculator gives the posterior P(A|B), the overall probability of B, P(A | not B), the likelihood ratio, and a natural-frequency table out of 10,000 people that makes the result easy to explain.

🧮 Formula

Bayes’ theorem
P(A|B) = P(B|A)·P(A) / P(B)
Total probability
P(B) = P(B|A)·P(A) + P(B|¬A)·P(¬A)

✏️ Worked examples

Rare condition, accurate test

You enter
Prior P(A)0.01
P(B | A)0.95
P(B | not A)0.05
You get
P(A | B) — posterior0.161017 (16.1017%)
P(B) — overall chance of B0.059 (5.9%)
P(A | not B)0.00053135 (0.053135%)
Likelihood ratio P(B|A) ÷ P(B|not A)19
In 10,000 cases95 true positives and 495 false positives → 16.1% of positives are real

0.161017 (16.1017%).

Spam filter

You enter
Prior P(A)0.4
P(B | A)0.9
P(B | not A)0.02
You get
P(A | B) — posterior0.967742 (96.7742%)
P(B) — overall chance of B0.372 (37.2%)
P(A | not B)0.0636943 (6.36943%)
Likelihood ratio P(B|A) ÷ P(B|not A)45
In 10,000 cases3,600 true positives and 120 false positives → 96.77% of positives are real

0.967742 (96.7742%).

🧭 How to use this calculator

What you enter

  • Prior P(A)e.g. how common a condition is: 1% = 0.01
  • P(B | A)test sensitivity / true-positive rate
  • P(B | not A)false-positive rate (1 − specificity)

What you get

  • P(A | B) — posterior
  • P(B) — overall chance of B
  • P(A | not B)
  • Likelihood ratio P(B|A) ÷ P(B|not A)
  • In 10,000 cases

❓ Frequently asked questions

What is a prior probability?

Your probability before seeing the evidence — for a disease, its prevalence in the group tested.

Why is the posterior so low for rare diseases?

Because even a small false-positive rate applied to the large healthy group creates more false positives than there are true cases.

Where is Bayes’ theorem used?

Medical diagnosis, spam filters, machine learning, forensic evidence, A/B testing and updating forecasts.

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