David Aldous asked me that question over lunch one day. I didn’t have an answer, so he explained: In the simplest model of human genetics, you get half your genes from each parent, a fourth from each grandparent, and so on. Thus the fraction of your genes contributed by each member of the nth generation is 1/2n. But there must be some value of n for which 2n exceeds the total number of genes in the human genome. Suppose you have 50,000 genes. Well, 16 generations ago you had 216 = 65,536 ancestors, so roughly 15,000 thousand of those family members were left out of the lottery. They’re your ancestors, but you inherited no genes from them.
There’s also a value of n for which 2n is greater than the entire human population, so if you look back far enough, you have more ancestors than there were people on the planet. This has got to be a sign of something awry in the model; these calculations are not to be taken as a quantitatively accurate guide to the human family tree. Nevertheless, the idea of counting genes and counting ancestors is basically sound.
These issues have come to the fore lately with news coverage of the discovery that Barack Obama is a distant cousin of Dick Cheney. According to Lynne Cheney, both are descended from Mareen Duvall, a 17th-century Hugenot. In today’s New York Times, Nicholas Wade comments on the significance (or otherwise) of this genealogical connection:
Mr. Obama probably inherited a minute fraction — one divided by two to the 11th power — of Mareen Duvall’s genome, which would amount to less than one gene, assuming the Y chromosome was not inherited.
Alas, though the concept is right, the numbers don’t quite add up. Two to the 11th power is only 2,048, and we surely have more genes than that. Under simple assumptions of random assortment, the expected number of genes passed down to the eleventh generation would be ten or so.
Wade correctly notes that the candidate and the vice president are very unlikely to have inherited any of the same genes from their common ancestor. Not that I would change my vote just because they had a few snippets of DNA in common.