# Bayes Theorem Calculator

Calculate posterior probability P(A given B) using prior, likelihood, and false-positive rate

> Canonical page: https://elysiatools.com/en/tools/bayes-theorem-calculator

- **Category:** Math & Numbers

- **Keywords:** bayes theorem, posterior probability, prior probability, likelihood, false positive, statistics

## Overview

Apply Bayes theorem.

Bayes theorem updates a prior probability after new evidence appears:

P(A | B) = P(B | A)P(A) / [P(B | A)P(A) + P(B | not A)P(not A)]

This is useful for diagnostic tests, spam filters, risk scoring, and evidence-based decision making.

## Inputs

- **Prior P(A)** (number)
- **Likelihood P(B given A)** (number)
- **False Positive P(B given not A)** (number)
- **Input Scale** (select)
- **Decimal Places** (number)

## When to use

- Evaluating the actual probability of having a condition after receiving a positive medical test result.
- Updating risk models or fraud detection algorithms when new behavioral evidence is observed.
- Solving statistics and probability problems that require applying Bayes' theorem to conditional probabilities.

## How it works

- Enter the prior probability (base rate) of the event occurring before any evidence is known.
- Input the likelihood (true positive rate) and the false-positive rate of the test or evidence.
- Select your preferred input scale (percentages or proportions) and set the desired decimal precision.
- The calculator applies Bayes' theorem to output the updated posterior probability as both a proportion and a percentage.

## Use cases

- Medical diagnostics to calculate the true chance of illness given a positive screening test and known disease prevalence.
- Machine learning and spam filtering to determine the probability that an email is spam given the presence of specific trigger words.
- Quality control in manufacturing to assess the likelihood that a flagged component is actually defective.

## Frequently asked questions

### What is prior probability?

Prior probability, or base rate, is the initial estimated likelihood of an event occurring before any new evidence or test results are considered.

### What does likelihood mean in this calculator?

Likelihood represents the true positive rate, or the probability that the evidence (such as a positive test result) appears when the event is actually true.

### Why is the posterior probability often lower than expected?

If the prior probability (base rate) is very low, even a highly accurate test with a small false-positive rate will yield a surprisingly low posterior probability. This statistical phenomenon is known as the base rate fallacy.

### Can I input values as decimals instead of percentages?

Yes, you can change the Input Scale setting from 'Percent' to 'Proportion' to enter decimal values like 0.01 instead of 1%.

### What is the false-positive rate?

The false-positive rate is the probability that the evidence or test result is positive even when the underlying event is false.

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