Yes, this is possible. The ABO blood types are determined by a single gene, and whether or not, someone is Rh- or Rh + is controlled independently by a different gene. Here is some more information:
We inherit 2 copies of our genes, one from our mother, and the other copy from our father. There are three different forms (called alleles) of the gene for blood type called A, B, and O. Blood type is determined by the combination of alleles inherited from our parents.
A person’s blood type is called their phenotype, and the different genetic combination of the forms of the gene is called a genotype. An example of a genotype would be OB. In this example, the person has two forms of the gene, O and B. This person’s blood type or phenotype would be type B blood. Another genotype for people with type B blood is BB.
Type A blood is another phenotype that can have the genotypes, AO or AA. In your example, if the mother’s genotype were AO, and the father had type B blood, then it would be possible to have a daughter with type B blood.
Whether blood is positive or negative is another phenotype. We all have 2 copies of the gene that determines if a blood type is positive or negative, and the following are genotypes: negative/negative; positive/negative, or positive/positive. Someone who is negative/negative has negative blood such as the father in your post. Someone who has positive blood can be either: positive/positive or positive/negative. The daughter in your example probably has the genoptype of positive/negative.
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