Bayes Theorem & Conditional Probability Calculator

Update prior probability P(A) with sensitivity P(B|A) and false-positive rate P(B|¬A).

Posterior Probability P(A | B)
38.462 %
Prior Probability P(A) 5 %
Total Evidence Probability P(B) 12.35 %
Positive Likelihood Ratio (LR+) 11.875 %
Inputs
%
Prior probability or base prevalence P(A) in percent.
%
True positive rate / sensitivity P(B|A) in percent.
%
False positive rate P(B|¬A) in percent.
Results & Graph
Posterior Probability P(A | B)
38.462 %
Prior Probability P(A)
5 %
Total Evidence Probability P(B)
12.35 %
Positive Likelihood Ratio (LR+)
11.875 %

Posterior Probability P(A | B)

Graph 1 • Prior Pct Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 % 26.5 % 53.0 % 50%: 23.342 % 23.3 50% 75%: 31.632 % 31.6 75% 100%: 38.462 % 38.5 100% 125%: 44.186 % 44.2 125% 150%: 49.053 % 49.1 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Posterior Probability P(A | B) 38.462 % Prior Probability P(A) 5 % Total Evidence Probability P(B) 12.35 % Positive Likelihood Ratio (LR+) 11.875 %
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Posterior Probability P(A | B)38.462 %Prior Probability P(A)5 %Total Evidence Probability P(B)12.35 %Positive Likelihood Ratio (LR+)11.875 %
Summary: Given a 5% prior, 95% sensitivity, and 8% false-positive rate, posterior probability P(A|B) is 38.46%.
Summary: Given a 5% prior, 95% sensitivity, and 8% false-positive rate, posterior probability P(A|B) is 38.46%.

Formula & Mathematical Methodology: Bayes Theorem Mathematical Identity

P(A|B) = P(B|A) × P(A)P(B|A)P(A) + P(B|¬A)(1 − P(A))
P(A|B) = [ P(B|A) × P(A) ] / [ P(B|A)P(A) + P(B|¬A)(1 − P(A)) ]
  • priorPct: Prior Pct (%)
  • sensitivityPct: Sensitivity Pct (%)
  • falsePositivePct: False Positive Pct (%)

Rounding: All calculations run in double-precision IEEE-754 arithmetic; displayed outputs round to up to 4 decimal places.

Worked Example: Worked Mathematical Verification: Bayes Theorem

Update prior probability P(A) with sensitivity P(B|A) and false-positive rate P(B|¬A).

  1. Verify the domain constraints of the input parameters.
  2. Apply the governing mathematical identity: P(A|B) = [ P(B|A) × P(A) ] / [ P(B|A)P(A) + P(B|¬A)(1 − P(A)) ].
  3. Cross-check the primary value against the supporting invariants and coordinate graph.
Result: Evaluates Bayes Theorem via P(A|B) = [ P(B|A) × P(A) ] / [ P(B|A)P(A) + P(B|¬A)(1 − P(A)) ].

Frequently Asked Questions

What mathematical formula does the Bayes Theorem & Conditional Probability Calculator use?

It directly evaluates P(A|B) = [ P(B|A) × P(A) ] / [ P(B|A)P(A) + P(B|¬A)(1 − P(A)) ] in double-precision floating-point arithmetic and updates all supporting invariants and graphs live.

How can I compare different parameter sets?

Click Compare in the results toolbar to pin up to three parameter combinations side by side.

Can I export the mathematical report or share a link?

Yes. Use the Share Link, Copy Summary, or Markdown/JSON export buttons in the results panel.

Mathematics & Statistics • Probability & Distributions Guide

How to Use the Bayes Theorem & Conditional Probability Calculator: Practical Guide & Visual Analysis

Calculate posterior probability P(A|B), total evidence probability P(B), and diagnostic positive/negative predictive values. Includes textbook mathematical notation, step-by-step verification, and three custom SVG visualizations. Whether you are researching bayes theorem & conditional probability calculator, bayes theorem, bayes theorem conditional probability, probability & distributions calculator, this interactive tool helps you calculate posterior probability P(A|B), total evidence probability P(B), and diagnostic positive/negative predictive values.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust Prior Pct, Sensitivity Pct, False Positive Pct, the calculator immediately computes Primary exact or numerical solution, Supporting identities & invariants, 3-Panel mathematical SVG plots and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Universal Mathematical & Statistical Standard (ISO 80000-2))

In our benchmark scenario (Worked Mathematical Verification: Bayes Theorem), Evaluates Bayes Theorem via P(A|B) = [ P(B|A) × P(A) ] / [ P(B|A)P(A) + P(B|¬A)(1 − P(A)) ]. Keep in mind that uses standard Euclidean geometry, real/complex algebra, and NIST statistical definitions. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Statistics & Standard Deviation Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder).