Final answer:
Both Jack and Jill must be carriers of the Tay-Sachs gene, meaning their genotypes are heterozygous (Tt). They each have one normal allele (T) and one Tay-Sachs allele (t), yet do not exhibit symptoms due to the presence of one functioning allele.
Step-by-step explanation:
Tay-Sachs disease is an autosomal recessive disorder, meaning that two copies of the defective gene must be present for an individual to express the disease. Since Jack and Jill's child has Tay-Sachs, we can infer the parents' genotypes. Both Jack and Jill must have at least one copy of the recessive gene (t), making them carriers of the disease. However, since they do not exhibit symptoms, they cannot have the genotype tt, which would result in them expressing the disease. Therefore, both Jack and Jill must be carriers, which means they are heterozygous for the Tay-Sachs gene, with genotypes of Tt.
In conclusion, Jack and Jill are both heterozygous (Tt) for the Tay-Sachs gene. Consequently, each parent has a 50% chance to pass the defective gene to their offspring. When both parents are carriers, there is a 25% chance that their child will inherit two recessive alleles (tt), leading to the expression of Tay-Sachs disease.