People - including experts - reason badly about conditional probabilities stated as percentages, because they neglect the base rate. Re-expressing the same facts as natural frequencies over a concrete population makes the correct answer nearly visible: "Out of 1,000, 10 have it; 9 of those test positive; of the 990 without it, ~89 also test positive; so of ~98 positives, only 9 truly have it - about 9%." The format does the work by keeping the base rate in the counts. The output is a natural-frequency breakdown. Honest constraint: the base rate and hit rates must be real - the format makes correct reasoning tractable, it does not invent the inputs.
When to Use
- Interpreting a test or screening result (medical, fraud, security, lead-scoring, A/B).
- Any "given a positive signal, what is the actual probability the thing is true?" question.
- Communicating risk to others so they do not over-read a positive.
When NOT to Use
- When you do not have real input rates and would have to invent them.
- When there is no conditional-probability structure to the question.
- For general project forecasting (use reference-class forecasting).
- When a single point estimate is wanted and the base-rate structure is irrelevant.
Instructions
When asked to reason about a conditional probability, follow these steps:
- State the question precisely. What posterior is being asked - usually P(condition | positive signal). Distinguish it from P(positive | condition), which people confuse it with.
- Gather the real inputs. The base rate, the true-positive (hit) rate, and the false-positive rate. If any is unknown, say so and stop or clearly flag the estimate as illustrative - do not fabricate numbers.
- Build a frequency tree over a concrete population. Pick a round number (e.g., 1,000). Work out: how many have the condition; of those, how many test positive; of those without, how many also test positive.
- Compute the posterior as true positives / all positives, and state it plainly.
- Name the wrong intuition it corrects. State the answer most people give (usually near the hit rate) and why it is wrong (base-rate neglect).
- Emit the natural-frequency breakdown per
references/TEMPLATE.md.
Output Format
Use the template in references/TEMPLATE.md. The deliverable is the frequency tree, the posterior, and the plain-language meaning, not a bare percentage.
Quality Checklist
Before finalizing, verify:
- The question distinguishes P(condition | positive) from P(positive | condition).
- The base rate, true-positive rate, and false-positive rate are real (or missing data is flagged, not invented).
- A frequency tree over a concrete population is shown.
- The posterior is computed as true positives / all positives.
- The common wrong intuition (base-rate neglect) is named.
- The output is the breakdown artifact, not a bare number.
Evidence
Tier S. Presenting conditional-probability information as natural frequencies substantially improves Bayesian-inference accuracy - accuracy on these problems rises from roughly 10% to 50-90% with the same facts in frequency format (Gigerenzer & Hoffrage 1995; Sedlmeier & Gigerenzer 2001), replicated across populations including physicians. The format does not supply the inputs; real rates are required. Evidence is from human reasoners, transferred to AI use, not AI-validated. Full grading: evidence/dossier.md.
Examples
See references/EXAMPLE.md for a completed breakdown.