Blood type is determined by what antigen (protein) type is found on the surface of red blood cells. Type A people have A antigens on their red blood cells, type B have B antigens, people with type AB blood have both, and those with type O have neither. Your question is about the genotypes involved. Type A individuals may have either AA or AO genotype. Type B can have BB or BO. Those with AB have received the allele for A from one parent and for B from the other (and their genotype is AB), and type O have received two copies of the allele for O (genotype OO).
Because the woman's children are type O and type B and the mother has type A, we know that the mother's genotype is AO; she has given the O gene to both children, as neither has A antigens. One child is type O, and therefore received the allele for O from both parents. We can therefore determine that one of the alleles in the father genotype is also for the O antigen. Because the other child is type B and received an O allele from the mother, the B must come from the father; the father is genotype BO, and he has type B blood.
In sum: mother is type A blood, genotype AO. Father is type B, genotype BO. Child 1 is type O, genotype OO. Child 2 is type B, genotype BO.
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