Final answer:
The most probable genotype for the mother with blood type A who has children with type O positive and type B negative blood is IAiRr, which satisfies the inheritance patterns for both ABO blood type and the Rh factor.
Step-by-step explanation:
The probable genotype for the mother in this case is IAiRr. Since the mother has blood type A and has a daughter with type O positive blood and a son with type B negative blood, she must possess both the A and O alleles as well as the Rh positive allele.
Blood type inheritance is determined by the ABO system and Rh factor. The IA and IB alleles are codominant, and the i allele is recessive. For the daughter to have type O (genotype ii), the mother must contribute an i allele, and the father must have an i allele as well. For the son to have type B, the mother must provide an allele for B; thus, her genotype related to ABO is IAi.
As for the Rh factor, since Rh positive is dominant over Rh negative and the mother must pass on the Rh positive allele to her Rh positive daughter, the mother's genotype must include an Rh positive allele (R). Since the son is Rh negative, the mother must also carry the Rh negative allele (r), giving her the Rh genotype of Rr. Combining the ABO and Rh genotypes gives us IAiRr as the mother's most likely genotype.