The short answer
Core relationship: Start with a fixed number of cases, apply the base rate, then apply sensitivity and false-positive rates to the relevant groups. Natural counts expose the comparison more clearly than isolated percentages.
For base-rate neglect in real decisions, begin with choose a concrete population and finish with compare flagged counts. The sequence keeps the starting facts, intermediate result, and final limit visible instead of relying on memory.
A reliable method
Choose a concrete population
Use 100, 1,000, or another count that keeps the rates readable. Capture the result before moving to apply the base rate first. A written output shows whether the next deduction follows from the evidence or quietly adds a new assumption.
Apply the base rate first
Separate target and non-target cases before considering the signal. Compare this output with the warning about quoting sensitivity alone. If that failure still fits, revise the representation before using the result in another deduction.
Apply each error rate
True positives come from one group and false positives from the other. Use apply each error rate as a checkpoint. Name the input, the change made, and the conclusion that becomes available so another reader can reproduce the move.
Compare flagged counts
The relevant question is what fraction of all flags are genuine. Finish by stating what this step establishes and what it leaves undecided. That boundary prevents a valid local conclusion from being stretched beyond the question.
Failure modes to watch
Quoting sensitivity alone
Detection among true cases does not answer truth among detections. Return to choose a concrete population and rebuild only from information the problem actually provides.
Forgetting the large group
A small false-positive rate can act on many non-target cases. Compare the disputed move with apply each error rate; the two should use the same definitions and direction.
Declaring the signal useless
A signal may substantially update probability without reaching certainty. Run the practice task while deliberately watching for declaring the signal useless. Record the first point where the reasoning begins to drift.
Worked case
Choose a concrete population in the case
If 1 in 100 items has a defect, a check that catches 90% of defects and falsely flags 5% of good items will flag about one true defect and five good items per 100. A flag is informative but not conclusive.
Why compare flagged counts matters
The case starts with choose a concrete population, then uses apply the base rate first to control the next inference. The conclusion is acceptable only after compare flagged counts. Changing any fact that supports those moves can change the answer, while an unrelated detail should not.
Check against quoting sensitivity alone
Questions for apply the base rate first
- Can another reader reproduce choose a concrete population from the stated information?
- Did you complete apply the base rate first without reversing or enlarging the claim?
- Does the result survive the failure mode quoting sensitivity alone?
- Which part remains uncertain after compare flagged counts?
This check is specific to the method above. A clear answer should expose the evidence chain and the point where a different fact would produce a different conclusion.
Try the method
Model 1,000 cases with a 2% base rate, 95% sensitivity, and 3% false-positive rate. Count true and false alerts before calculating a percentage.
Continue after compare flagged counts
Use the probability reasoning guide for the broader method, then apply the idea in the probability calculator. Keep the written reasoning so the next attempt can improve a specific step rather than simply repeat the exercise.
Compare this method with Conditional Probability Without Formula Confusion, then continue to Expected Value for Everyday Decisions for a different reasoning task.