Final answer:
With the mother having the genotype rr (homozygous recessive) and the father having the genotype Rr (heterozygous), giving them a 25% chance of having another red-haired child. The correct option is b.
Step-by-step explanation:
When considering the genotypes of the parents and the likelihood of their children having a particular trait such as red hair, it is essential to understand Mendelian inheritance. Hair color is typically determined by multiple genes, but to simplify the situation, let's assume a model where red hair is controlled by one gene with two alleles: the red hair allele (r) being recessive and the non-red allele (R) being dominant.
The mother has red hair, which means she must have two recessive alleles (rr). The father has brown hair but they have a child with red hair, indicating that he carries at least one recessive allele, so his genotype must include r (making his genotype Rr if he carries both brown and red alleles).
With the information given, we can refute options (a), (c), and (d), because they imply that the father does not carry the recessive allele, or that the mother isn't homozygous for red hair. Therefore, the correct answer would be option (b): Mother: rr, Father: Rr; with a 25% chance of them having another child with red hair.
This is because the expected ratio of offspring from an Rr and rr cross is 1:1:1:1 for the genotypes RR, Rr, rR, and rr. Thus, only one in four offspring is expected to be homozygous recessive (rr) with red hair. Hence, b is the correct option.