Mathematics · Probability
Bayes Theorem Calculator
Update a prior probability after a positive observation.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- True-positive share=0.01×0.9=0.009000000000000001.
- False-positive share=0.99×0.05=0.0495.
- Posterior=0.009000000000000001÷0.0585=0.15384615384615385.
Problem → model → reason → result
What problem does this model solve?
Update a prior probability after a positive observation.
Why does the model apply?
Bayes' theorem balances evidence likelihood against the prior base rate.
What assumptions does it make?
Outcomes and selection rules match the stated model; independence, equal likelihood and replacement are assumed only when the problem actually provides them.
Formula
P(A|+)=sensitivity·prior/[sensitivity·prior+false-positive·(1−prior)]
Calculation and working
The calculator above substitutes your inputs into the model and leaves the calculation steps visible so the answer can be checked rather than merely accepted.
What does the result mean?
The result answers the stated problem in the units implied by your inputs. Read its sign, size and units together, then compare it with the original values before drawing a conclusion.
Worked example
Prior 1%, sensitivity 90%, false-positive 5% gives posterior about 15.4%.
Common mistake
A highly accurate test can still have a modest posterior when the condition is rare.
When does this model not apply?
The numerical answer is only as valid as the assumed outcome space and dependence structure.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Bayes theorem do?
Update a prior probability after a positive observation.
How does the Bayes theorem work?
The calculator applies P(A|+)=sensitivity·prior/[sensitivity·prior+false-positive·(1−prior)]. Bayes' theorem balances evidence likelihood against the prior base rate.
What can I learn from the Bayes theorem?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.