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.
Understand Bayes theorem
One idea, three depths
Choose how deeply to explain Bayes theorem
Bayes theorem: Update a prior probability after a positive observation.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Bayes theorem to answer this question: update a prior probability after a positive observation? Enter Prior P(A), Sensitivity P(+|A), False positive P(+|not A); the calculator shows Posterior P(A | +). For example: Prior 1%, sensitivity 90%, false-positive 5% gives posterior about 15.4%. The answer tells you Posterior P(A | +).
Age 15Explain it to a 15-year-oldConnect it to the formula
Bayes' theorem balances evidence likelihood against the prior base rate. The rule is P(A|+)=sensitivity·prior/[sensitivity·prior+false-positive·(1−prior)]. Its input values are Prior P(A), Sensitivity P(+|A), False positive P(+|not A), and the main result is Posterior P(A | +). For example: Prior 1%, sensitivity 90%, false-positive 5% gives posterior about 15.4%.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated bayes theorem relation over the valid real-number domain stated below. The implemented relation is P(A|+)=sensitivity·prior/[sensitivity·prior+false-positive·(1−prior)], evaluated from Prior P(A), Sensitivity P(+|A), False positive P(+|not A) to produce Posterior P(A | +). Bayes' theorem balances evidence likelihood against the prior base rate. A highly accurate test can still have a modest posterior when the condition is rare.
Inputs and valid domain
- Prior P(A) must be a finite real number, at least 0, at most 1.
- Sensitivity P(+|A) must be a finite real number, at least 0, at most 1.
- False positive P(+|not A) must be a finite real number, at least 0, at most 1.
Important boundary: A highly accurate test can still have a modest posterior when the condition is rare.
The formula
P(A|+)=sensitivity·prior/[sensitivity·prior+false-positive·(1−prior)]
How the calculator works through it
It substitutes Prior P(A), Sensitivity P(+|A), False positive P(+|not A) into the formula and exposes every numerical step above. The main output is Posterior P(A | +), accompanied by Posterior percent, Positive evidence probability.
Read the result correctly
The Posterior P(A | +) is the direct answer to “update a prior probability after a positive observation.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
Prior 1%, sensitivity 90%, false-positive 5% gives posterior about 15.4%.
Where this model stops being reliable
A highly accurate test can still have a modest posterior when the condition is rare.
Learn it by changing one value
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.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Bayes theorem works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Bayes theorem uses P(A|+)=sensitivity·prior/[sensitivity·prior+false-positive·(1−prior)]. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Probability as a modelled proportion
Probability rules are needed to interpret what the Bayes theorem result says about possible outcomes.
Review this foundation about 5 min
Optional enrichment
- Ordered arrangements
Counting ordered arrangements can extend Bayes theorem to more detailed sample spaces and event models.
Review this foundation about 5 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Prior P(A), Sensitivity P(+|A), False positive P(+|not A).
- Evaluate the principal relationship: P(A|+)=sensitivity·prior/[sensitivity·prior+false-positive·(1−prior)].
- Return Posterior P(A | +) and check the domain conditions described above.
Python
from math import *
def bayes_theorem(a, b, c) -> float:
return ((b * a) / ((b * a) + (c * (1.0 - a))))
assert abs(bayes_theorem(0.01, 0.9, 0.05) - 0.15384615384615385) < 1e-6 * max(1.0, abs(0.15384615384615385))
C
#include <assert.h>
#include <math.h>
double bayes_theorem(double a, double b, double c) {
return ((b * a) / ((b * a) + (c * (1.0 - a))));
}
int main(void) {
const double expected = 0.15384615384615385;
const double actual = bayes_theorem(0.01, 0.9, 0.05);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double bayes_theorem(double a, double b, double c) {
return ((b * a) / ((b * a) + (c * (1.0 - a))));
}
int main() {
constexpr double expected = 0.15384615384615385;
const double actual = bayes_theorem(0.01, 0.9, 0.05);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double bayes_theorem(double a, double b, double c)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global bayes_theorem
section .text
bayes_theorem:
push rbp
mov rbp, rsp
sub rsp, 80
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-8]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-8]
movsd [rbp-56], xmm0
mov rax, 0x3ff0000000000000
movq xmm0, rax
movsd [rbp-80], xmm0
movsd xmm0, [rbp-80]
subsd xmm0, [rbp-8]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-24]
mulsd xmm0, [rbp-72]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-56]
addsd xmm0, [rbp-64]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-40]
divsd xmm0, [rbp-48]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = bayes_theorem(a, b, c)
result = ((b * a) / ((b * a) + (c * (1.0 - a))));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, c_] := ((b * a) / ((b * a) + (c * (1.0 - a))));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Introductory Statistics 2e
Read the free OpenStax statistics textbookCite this book
- APA 7
- Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
- MLA 9
- Illowsky, Barbara, and Susan Dean. Introductory Statistics 2e. OpenStax, 2023, https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
- Chicago author-date
- Illowsky, Barbara, and Susan Dean. 2023. Introductory Statistics 2e. Houston, TX: OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Bayes Theorem Calculator. MW SysArc Tools. https://math.mwsysarc.com/probability/bayes-theorem
MLA 9
MW SysArc. “Bayes Theorem Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/probability/bayes-theorem. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Bayes Theorem Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/probability/bayes-theorem.
Harvard
MW SysArc (2026) ‘Bayes Theorem Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/probability/bayes-theorem (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_bayes_theorem_2026,
author = {{MW SysArc}},
title = {Bayes Theorem Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/probability/bayes-theorem},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Bayes Theorem Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/probability/bayes-theorem
N1 - Published July 21, 2026
ER -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 . Calculations tested .