Calculating Bayes’ Theorem
Multiply likelihood by prior to obtain the joint probability, then divide by the total evidence probability.
Math calculator
Bayes’ Theorem Calculator
Update a prior using a likelihood and total evidence probability. The calculation trail makes the reported posterior probability easier to reproduce.
Bayes’ Theorem inputs
Values for this result
Example from start to finish
Prior 0.10, likelihood 0.80, and evidence 0.17 give posterior 0.08/0.17≈0.4706. This Bayes’ Theorem example can be compared with conditional definition.
When the result is useful
Bayesian updates appear in screening tests, classification, diagnosis, quality control, and forecasting.
Reading the calculation
Bayes’ theorem states P(A|B)=P(B|A)P(A)/P(B). It reweights a prior by compatibility with evidence and normalizes across every route to that evidence.
A correct posterior probability therefore depends on choosing the model before entering the numbers.
Boundaries and common traps
Evidence probability must include every mutually exclusive cause of B. Confusing P(B|A) with P(A|B) creates a base-rate error. If the Bayes’ Theorem assumptions do not fit, consider alternative state.
Review the sign, scale, and unit of posterior probability after entering prior p(a), likelihood p(b|a) and evidence probability p(b).
From calculation to usable answer
Source values may arrive in a different order from the form. Map them explicitly to prior p(a), likelihood p(b|a) and evidence probability p(b), normalize units, and retain enough precision for the next step after posterior probability.
A strong likelihood can raise a posterior substantially, but a rare prior may still keep it moderate. Watching this response separates a data-entry mistake from an unexpected but valid value.
If prior p(a), likelihood p(b|a) and evidence probability p(b) are exact counts, more result digits may be meaningful than when they are measured approximations. Let the least certain source guide the final presentation.
Conditional probability defines the ratio; Bayes reorganizes information to reverse the condition. This page deliberately reports only the former interpretation.
Label the output as posterior probability in notes or tables. Store prior p(a), likelihood p(b|a) and evidence probability p(b) beside it when the result will be reused in a later stage.
Assumptions behind Bayes’ Theorem
The Bayes’ Theorem formula describes a particular experiment built from Prior P(A), Likelihood P(B|A), Evidence probability P(B). Write down whether the trial count is fixed, objects return after selection, events overlap, or outcomes are equally likely. Similar-looking numerical inputs can require different formulas when one assumption changes.
For Bayes’ Theorem, after calculation, translate the answer back into a sentence about the original event. That wording should distinguish exactly, at most, at least, all, and none; substituting one of those phrases for another changes the event rather than its formatting.
Questions about Bayes’ Theorem
What is the prior?
Probability before observing B.
What is the posterior?
Updated probability after B.
Why normalize by evidence?
It includes every route to B.
Can a positive result imply less than 50%?
Yes.