Before calculating, name the event you are trying to estimate. "Carries the variant," "has the condition," and "has a negative result" are different events. Many errors begin when a correct number is attached to the wrong question.
Multiply independent steps for a joint event
Two unaffected partners have carrier probabilities of 1/5 and 1/20 for the same fully penetrant autosomal recessive condition. Model each partner as AA or Aa and assume their carrier risks are independent. The probability of an affected child is:
1/5 × 1/20 × 1/4 = 1/400 per pregnancy
The first two factors estimate the chance both are carriers; the last estimates an affected child if both are carriers. Relatedness or other shared information can make an independence assumption inappropriate.
Update the prior after a negative test
For an illustrative assay, assume a prior carrier probability of 1/20, sensitivity of 95%, and specificity of 100%. A negative result can occur in a carrier whom the assay misses or in a noncarrier.
- Carrier and negative: 1/20 × 1/20 = 1/400
- Noncarrier and negative: 19/20 × 1 = 380/400
- Residual carrier probability among negative results: 1/(1 + 380) = 1/381
The 5% false-negative rate is not the residual carrier probability. The posterior also depends on how likely carrier status was before testing. These assay values are hypothetical and should not be transferred to a real laboratory test.
| Branch | Probability | What it represents |
|---|---|---|
| Carrier and negative | 1/20 × 1/20 = 1/400 | Carriers whom the assay missed |
| Noncarrier and negative | 19/20 × 1 = 380/400 | True noncarriers with a negative result |
| Posterior among negatives | 1 ÷ (1 + 380) = 1/381 | Carrier branch divided by the total negative branch |
Use a general expression when helpful
Posterior carrier probability after a negative result:
[Prior × (1 − sensitivity)] ÷ {[Prior × (1 − sensitivity)] + [(1 − prior) × specificity]}
Use this only when the stated test model fits the problem. Real residual-risk estimates may depend on the condition, assay, ancestry data, and laboratory reporting conventions.
Practice the setup before increasing speed
Label each branch before entering numbers. After solving, ask whether the result is plausible: under the model above, a negative result should lower carrier risk without making it zero. Use the free Genetics Math Workbook for written practice and EveryGene Academy for case-based application.
See also: Hardy-Weinberg practice problems and pedigree practice problems.
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