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StatisticsBayesian update

Bayes Theorem Calculator

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Update a prior probability into a posterior probability after observing evidence. Use diagnostic mode for sensitivity/specificity or direct mode for general conditional probabilities.

This page is useful when you want to explain how prior belief, evidence strength, and Bayes factor work together before you jump to likelihood-ratio shorthand.

How to use

  1. Choose diagnostic mode when you already have sensitivity and specificity for a positive or negative result.
  2. Choose direct mode when you want to enter P(E|H) and P(E|not H) directly.
  3. Read posterior probability together with evidence probability and Bayes factor so the update stays interpretable.

Wave 9 probability bridge

Prior → evidence → posterior

The share URL stores only mode, result choice, and decimal places. It does not include the probabilities you enter.

Diagnostic inputs

In diagnostic mode, positive evidence uses sensitivity and false-positive rate. Negative evidence uses false-negative rate and specificity.

Choose a mode, enter the probabilities, and run the calculation.

When to use Bayes theorem directly

Use this page when you need to show the structure of the update itself: prior probability, evidence under the hypothesis, evidence under the alternative, and the final posterior. That is often the clearest teaching path before you switch to LR+ or LR−.

If your workflow already starts from LR+ or LR−, move to pre-test / post-test probability. If you still need the evidence ratios, check likelihood ratio or NPV & PPV.

Frequently asked questions

What does this Bayes theorem page calculate?

It updates a prior probability into a posterior probability after you observe evidence. You can enter direct conditional probabilities or use sensitivity and specificity for a diagnostic-style example.

How is this different from the pre-test / post-test probability page?

This page explains the probability update directly from Bayes' theorem. The pre-test / post-test page starts from likelihood ratios, which is often the faster workflow after you already know LR+ or LR−.

What is a Bayes factor here?

The Bayes factor is the evidence ratio P(E|H) divided by P(E|not H). Values above 1 support the hypothesis, while values below 1 support the alternative.

Does the share URL include my inputs?

No. The share URL stores only lightweight settings such as mode, result, and decimal places. The probabilities you enter stay in your browser.