Is it possible for the father to be blood type O and the mother to be blood type A and have a child with blood type B? If so, how?

Blood Types Tutorial (http://www.biology.arizona.edu/human_bio/problem_sets/blood_types/Intro.html)

Blood Types Tutorial (http://www.biology.arizona.edu/human_bio/problem_sets/blood_types/genotypes.html)

Yes, it is possible.
There is a rare blood type called hh (Bombay blood group) which prevent the expression of the A & B antigens. So a person who has the A and/or B alleles (genotype) will have the blood type O phenotype. See the Hh antigen system at Wikipedia (hh antigen system - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Hh_antigen_system)) for the details.

Possible…. But how frequently does this happen?
Well it depends. In Europeans it’s is very rare (one in a million), but more common in other places. In India (remember it’s the Bombay phenotype) it is 100X more common & on Reunion Island (Réunion - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Reunion_Island)) the frequency is one in a thousand.

The genetic details for the H/h blood group are a little more completed (two homologous, linked genes (FUT1 & FUT2) with different tissue specific gene expression patterns). Check out dbRBC - Blood Group Antigen Gene Mutation Database (http://www.ncbi.nlm.nih.gov/gv/mhc/xslcgi.cgi?md=bgmut/systems_info&system=hh) for the details.