You're using a pretty racist rule that's the "one drop" rule of determining someone's race/ethnicity where other races perpetuate and "take over" as soon as there's "one drop of non white blood in the lineage" while whitness has to be "preserved".
Generally once someone is mixed race it's a genetic crapshoot as to how they look, so if they look more like one parent than another a lot of how they define their ethnicity will also be how they're raised and what part of their parent's cultures they resonate with at different times, it can get really complicated trying to work out how to juggle both.
That also means that say if someone is 1/8th black and 7/8ths white, because their grandfather had a black dad but everyone else in their lineage has been caucasian, and they no longer have any ties to black ancestry, culture, art, common shared experiences, etc... and look white, they are treated as white, they might even consider themself white, or not even know there's anything other than white in their ancestry. They are, for all intents and purposes, white, despite being technically "mixed".
I say this as a very pale english white person: by the one drop rule, I am black, and that is a ridiculous statement to me because the cultural ties to this great great great grandfather were lost to family history and had to be dug up in archives. It's not even that far back in the lineage that our family still looked black or at least not entirely white, we just didn't know our lineage prior to our grandmother because of adoption, yet it would be disrespectful of me to claim I have any personal insight into what it's like to be any other ethnicity than caucasian, so I base my observations on my friends' experience as they tell it, on the kids I work with, and how I can somewhat sympathise with the living in two cultures simultaneously because though not being mixed race, i am a dual citizen, though it's far from the same as being observably different.
I think you should reconsider your conception of race, it seems very archaic. This is a very complex subject that interacts with many other facets of life in sometimes unexpected ways, and can very easily be insulting to many people, so it's important to handle them with care.