Well. We would need to make the conditions precise to get a definite answer.

If

- your fraction of being of a given ethnicity is the average of the corresponding fractions of your parents, and
- originally (a finite number of generations ago) all the people were ethnically pure,

then the conclusion is that the denominator of your fraction is a power of two.

Proof by induction on the number of generations you need to trace back to have only ethnically pure ancestors. I can give more details, but I'm not sure you want me to. The crude picture is that by induction hypothesis your parents have denominators that are powers of two. If their powers are different, then your denominator will be twice the bigger one of your parents. If their powers of two are the same, then yours is the same or possibly smaller. The base case was included in the assumptions.

Notice that under these assumptions inbreeding won't change anything. For example if your parents were cousins, the ethnicity of their common pair of grandparents will have twice the weight in comparison to their other grandparents.

If you have different rules for counting your percentage of being of nationality X, then the conclusion may, of course, be different.

This gedankenexperiment is fun, but I do question the assumption of purity of the distant ancestors. How might we define that?