What fraction of a parent’s genes exist in total across n offspring? This problem has perplexed me for a long time, and I would love to learn how to solve it.


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*Each parent contributes, on average, half of the genetic material of each child. If a parent has $n$ children, what percentage of the parent’s genes exist in total across all $n$ offspring?

*How can one generalize the solution to #1, to make it applicable to more distant relationships like uncles & aunts, grandparents, and cousins, where the fraction of shared genes is smaller?
For the purposes of these questions, disregard the complexities of chromosomes, meiosis, and fertilization, and just assume that genes are chosen from each parent independently and at random.
 A: We need a few more details of human genetics before answering this question,
For a single gene, of which each person has two copies, there are different versions of that gene within the population, these are called alleles. A person can have two copies of the same allele (homozygous) or each copy may be a different allele (heterozygous). In humans, parents pass on one allele of each gene to their offspring.
For a gene that the parent is homozygous for, it is simple, all offspring inherit this allele from the parent. 100% of homozygous genes are passed on.
For a gene that a parent is heterozygous for, we need to work out the probability that at least two offspring have different alleles of the gene (let us call the two alleles $H$ and $h$). The only way this can not happen is for all offspring to have $H$ or all to have $h$.
$P(H \space\& \space h\space  present) =  1 - (P(all \space H) + P(all\space h))$
Since there is only one way to get all $H$ and one way to get all $h$ the calculation is straightforward for $n$ children
$ P(H \space\& \space h\space  present) =   1 - (0.5^{n} + 0.5^{n})$
With $n=6$ you can be more than 95% sure that both alleles of a given gene are present in the children. So, out of 100 heterozygous genes and 6 children, for ~95 of them, both alleles will be present in the children, and for the remaining ~5 only one allele will be present.
The total genetic information passed to offspring depends on how likely a particular gene is to be homozygous or heterozygous... which in turn depends on the relative frequencies of the alleles for a particular gene in the population... this will also affect how likely both parents are to have the same allele, a factor which we ignored in the calculation above. 
As you can see it gets complicated quite quickly, so you would probably have to make some assumptions about the distribution of allele frequencies in the population.
Interestingly the Hardy-Weinberg principle states that the frequencies of alleles in a population remain roughly constant. So you can be sure, under the assumptions of that model, that all your alleles are preserved somewhere across the population!
