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Natural-Frequency Bayesian Framing

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:

1. 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.
2. 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.
3. 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.
4. Compute the posterior as true positives / all positives, and state it plainly.
5. 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).
6. 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 - in the original study, correct Bayesian answers rose from 16% to 46% with the same facts in frequency format (Gigerenzer & Hoffrage 1995), and a meta-analysis puts the average at about 4% versus 24% (McDowell & Jacobs 2017). Most people still answer wrongly even with natural frequencies, so the format helps without guaranteeing a correct answer. Replicated across populations including physicians. The format does not supply the inputs; real rates are required. For two sequential tests, one preregistered trial found the advantage reverses: odds and likelihood ratios did better (Schulz, Wegwarth & Giese 2025). 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.