FINAL TOOLBOX
01 // Chart Studio02 // Visual Solvers03 // Calculators04 // Formulas05 // Unicode & Symbols
03 // Bidirectional STEM Calculators // STATISTICSP(A|B) = (P(B|A) × P(A)) / P(B)

Bayes' Theorem Posterior Probability Calculator

Bayes' Theorem updates the prior belief probability P(A) of a hypothesis given new diagnostic or experimental evidence B, scaling by the likelihood ratio P(B|A) / P(B).

P(A|B) = (P(B|A) × P(A)) / P(B)
Pick which value you want to calculate (or type into any field):Solving for: P(A|B) — Posterior Probability
Calculated

Updated probability of A given evidence B occurred

0.00010.9999
unit

Probability of observing evidence B when A is true

0.0010.9999
unit

Baseline prevalence or prior probability of A

0.00010.9999
unit

Total probability of observing B across all states

0.00010.9999
unit
Calculated Result · Posterior Probability (P(A|B))
0.475

Step-by-Step Calculation

01Canonical Law: P(A|B) = (P(B|A) × P(A)) / P(B)
02Given Inputs (Base Units): P(B|A) = 0.95, P(A) = 0.05, P(B) = 0.1
03Solve for P(A|B) (Posterior Probability): P(A|B) = 0.475

Bayes' Theorem updates the prior belief probability P(A) of a hypothesis given new diagnostic or experimental evidence B, scaling by the likelihood ratio P(B|A) / P(B).

Real-World Reference Values

Bayes Factor / Likelihood RatioP(B|A) / P(B) = 9.50×

Multiplier updating prior P(A) to posterior P(A|B)

Base-Rate Fallacy Illustration5% Prior + 95% Sensitivity → 47.5%

False positives dominate when base rate P(A) is low

Law of Total ProbabilityP(B) = P(B|A)P(A) + P(B|¬A)P(¬A)

Decomposes marginal evidence into true + false positives

20-Row Comparison Table

Posterior Probability (P(A|B)) · Likelihood / Sensitivity (P(B|A))

Bayes' Theorem Posterior Probability Calculator — Likelihood / Sensitivity → Posterior Probability
#Likelihood / Sensitivity (P(B|A))Posterior Probability (P(A|B))Δ %
010.3515 0.17575 -63.00%
020.411939 0.20597 -56.64%
030.472379 0.236189 -50.28%
040.532818 0.266409 -43.91%
050.593258 0.296629 -37.55%
060.653697 0.326849 -31.19%
070.714137 0.357068 -24.83%
080.774576 0.387288 -18.47%
090.835016 0.417508 -12.10%
100.895455 0.447728 -5.74%
110.95 Current0.475 0.00%
121.0163 0.508167 +6.98%
131.0768 0.538387 +13.34%
141.1372 0.568607 +19.71%
151.1977 0.598826 +26.07%
161.2581 0.629046 +32.43%
171.3185 0.659266 +38.79%
181.379 0.689486 +45.15%
191.4394 0.719705 +51.52%
201.4999 0.749925 +57.88%
Was this tool helpful?

Frequently Asked Questions

Why can a 95% accurate test have only a 47.5% Positive Predictive Value?+

When a condition is rare (e.g., P(A) = 5% prevalence), false positives from the healthy 95% of the population rival true positives in the 5% affected group.

How do I find the prior prevalence P(A) required to achieve 90% posterior confidence?+

Click SOLVE FOR on P(A) and set P(A|B) = 0.90.

Can Bayes' Theorem be applied sequentially?+

Yes. Today's posterior probability P(A|B₁) becomes tomorrow's prior P(A) when incorporating a second independent test B₂.