Two evidence folders and connected wooden tokens on a quiet desk. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Odds make one update compact
For hypotheses A and not A, prior odds are P(A)/P(not A). Evidence B contributes the likelihood ratio P(B | A)/P(B | not A). Posterior odds equal prior odds multiplied by that likelihood ratio. Convert the final odds o back to a probability with o/(1+o). This odds statement and the usual Bayes fraction describe the same update when the denominators are defined.
Work one inspection example
A prior probability of 10% gives odds 1:9. A flag with likelihoods 80% under A and 20% under not A has likelihood ratio 4. Posterior odds become 4:9, corresponding to 4/13 or about 30.7692%. A second equally informative flag would produce odds 16:9 and probability 64% only if the evidence model justifies the second factor of four.
Why repeated evidence may add much less information
The second update needs P(B₂ | A, B₁) divided by P(B₂ | not A, B₁). Those conditional likelihoods equal the original marginal likelihoods only under suitable conditional independence. Running the same image through the same deterministic model twice does not provide two independent inspection flags. Two measurements affected by the same contamination or sensor fault may also share information. Multiplying marginal likelihood ratios without checking the process can make the result far too confident.
Write the evidence protocol beside the numbers
Record how each result was obtained, what earlier information was already used in the prior, and whether the likelihood model accounts for dependence. Avoid counting the same observation once in the prior and again as new evidence. The Bayes' Theorem Calculator supports one binary-evidence update; sequential use is appropriate when you supply the correct conditional likelihoods at each step. Read the supplier inspection example for a direct probability-table calculation. If you only need to reverse a conditional from a known joint table, the Conditional Probability Calculator is the more direct tool.
Check the model behind your result
Use the Bayes' Theorem Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26