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If I'm type TU, what type are my kids? The first thing to understand is that each of us has two blood type genes. Each parent can pass only one of his or her genes to each child; and if we have two different genes -- as in the AB type -- it can be either of those genes.
Second, when we say, "I'm type B," or "He's type AB," we are referring to what geneticists call the phenotype. It's the outward face of the genotype, the full ABO signature (always composed of two genes).
Third, the O gene is recessive to both the A and the B genes. The other way of saying this is, both the A and the B gene are dominant to the O gene. I am "type A" or "type B" whether I carry a second (recessive) O gene or a matching A or B gene.
Fourth, the A and B genes are co-dominant: when both are present, neither is recessive to the other. If both are in the genotype (full profile: A, then both are in the phenotype (blood type A. In similar fashion, if you are type O, we know you have two O genes ~ if you had only one O gene and one A or B gene, the A or B must dominate: you would be type A or type B, respectively, not type O.
This all sounds a little confusing at first, but you'll get the hang of it. Here's how it works in parentage:
Type AB can pass on either an A or a B gene to a particular child -- but not both.
Type O has only an O gene to contribute to a child.
Type A can give an A gene to a child. However, we can't tell a type A person's genotype by knowing their phenotype. The genotype could be A(a)(with only A genes to give to a child) or A(o) (having an O gene available as well).
Type B is in the same relative situation as type A. There could be a recessive O gene lurking in there (Peter calls it "grandma's revenge!") ready to match up with a mate's hidden O gene, to produce a type O individual from two type B parents!
(1) that each person contributes one gene to offspring;
(2) that A and B are both dominant to O, and co-dominant with each other; and
(3) that knowing the "phenotype" of an AB or O person lets us know their "genotype," but knowing someone is type A or B gives us only the "public" half of that information -- the second gene is unknown;
will allow you to figure out the possible blood types of offspring if you know the two parents' blood types. With some combinations, there is only one possible result (two type O parents will always produce type O kids). With others, it's possible that Junior could be any one of the four types (a type A mom and type B dad, both with a recessive O gene... see what I mean?). In your original example, it depends on the other parent. Type O? This pairing will produce only type A (A(o)) and B (B(o)) children. Type AB? Could be type As, type Bs and/or type ABs.
The most elegant way to present this information is in graph form (a "Punnett square"), but I hope yakking my way through it has shed some light for you.