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Author Topic: Genetics and fractions  (Read 2477 times)

Pilchard123

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Genetics and fractions
« on: 15 Aug 2013, 10:35 »

If the only restriction is that there may not be an infinite number of generations, is it possible to be 1/n parts (nationality), where n is any odd integer? What if n is any prime?

If not, why not?
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cesium133

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Re: Genetics and fractions
« Reply #1 on: 15 Aug 2013, 10:54 »

It depends. Is inbreeding allowed? If so, then yes. If not, then no. I'm going to assume it's not allowed (though in any realistic situation it must occur: there simply weren't enough humans in the past for there not to be).

If we assume the family tree uniformly branches such that each person has two parents, unique from any other people on the tree, then if we consider a set on level N of the tree, there are 2^N members on that level. Therefore any subset must be a fraction with denominator 2^N. It would be possible to approximate a fraction with an odd denominator, but not equal it exactly, as long as N is finite. If N is allowed to be infinite (although that is not allowed in your description), then it is possible to get a fraction with an odd denominator.
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Pilchard123

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Re: Genetics and fractions
« Reply #2 on: 15 Aug 2013, 14:20 »

Yes, inbreeding is allowed. I suppose the question would have been better had I phrased it more generally rather than as family trees. That's the answer I thought was the case, though I'm trying to find a way to prove it.
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snalin

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Re: Genetics and fractions
« Reply #3 on: 27 Aug 2013, 03:05 »

Homework thread!
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Re: Genetics and fractions
« Reply #4 on: 29 Aug 2013, 13:45 »

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?
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