A Negative Carrier Test and the Brochure's Promise
Alex, 16, has just learned that his cousin - the son of his father's sister - has a condition inherited as an autosomal recessive trait. The family tree shows Alex's paternal grandparents had four children, all unaffected: Alex's father and three aunts, one of whom is the cousin's mother. Alex's mother has no family history of the condition; the population carrier frequency is estimated at 1 in 25. Alex's father takes a carrier test that screens only the most common variants of the gene; the brochure states it detects about 90% of carriers. The result is negative. The same brochure claims: "A negative result means your children will not be affected." Alex asks his science teacher whether he should be worried and whether the test settles the question; she asks him to work out the probabilities himself before answering.
1. What is the probability that Alex's father carries the variant? Explain why the answer is not 2/3.
Read the worked answer
Start from the cousin. He is affected by a recessive condition, so he inherited one variant from each parent. His mother - Alex's aunt - is unaffected, which means she carries exactly one copy of the variant.
That copy came from one of Alex's paternal grandparents, so at least one grandparent is a carrier. This is the key point: we know only ONE grandparent's status. Alex's father, an unaffected child of that grandparent, received one of that grandparent's two alleles at random, so P(father is a carrier) = 1/2.
The 2/3 figure that students often reach for applies only when BOTH parents are known carriers and the child is known unaffected - for example, the aunt's own children. Using 2/3 here would assume both grandparents are carriers, which the family tree does not show. (The second grandparent could carry the variant at the population rate of 1/25, but that barely changes the result.)
2. Before the test, what was the probability that Alex himself is a carrier, and what would that mean for a future child?
Read the worked answer
Two independent steps are needed:
- P(father is a carrier) = 1/2, from the aunt's affected son;
- if the father is a carrier, each of his children has a 1/2 chance of inheriting the variant.
So P(Alex is a carrier) = 1/2 × 1/2 = 1/4, or 25%.
Now add a partner drawn from the general population, whose carrier probability is 1/25. For the couple to have an affected child, all three events must happen:
P = 1/4 × 1/25 × 1/4 = 1/400 ≈ 0.25%
The last 1/4 is the chance that both parents pass on their variant. Note what drives the number: the family history raises Alex's risk far above the population baseline, but the child's risk is still small because the partner is very unlikely to be a carrier. Also, the 1 in 25 figure is a population average - it would not apply if the partner came from a group with a higher frequency.
3. Recompute Alex's carrier probability after the negative test, then evaluate the brochure's claim.
Read the worked answer
The test is not perfect. If it detects about 90% of carriers, then roughly 10% of carriers test negative. Alex's father had a 1-in-2 chance of carrying the variant, so the residual chance that he is a carrier despite the negative result is about 1/2 × 1/10 = 1/20. (A formal Bayesian calculation, which accounts for the test being negative in non-carriers too, gives a slightly higher value, near 1/11.)
Alex's own probability therefore falls from 1/4 to about 1/2 × 1/20 = 1/40, roughly 2.5%. With a population partner the child's risk becomes about 1/40 × 1/25 × 1/4 ≈ 1/4 000 - small, but not zero.
The brochure is wrong on two counts:
- A screen for common variants cannot detect variants it does not look for, so a negative result lowers the probability without eliminating it.
- "Your children will not be affected" ignores the partner entirely. A child's genotype depends on both parents; no test on one parent can guarantee a child's outcome.
A realistic recommendation: the risk is low, but a genetic counsellor can interpret the specific test and family history.