You have one pair of parents, 2 pairs of grandparents (whose other grandkids are your first cousins), 4 pairs of great-grandparents, and so on. A 10th cousin will share parents 11 generations back -- where there's a stock of 1024 pairs of ggggggggggrandparents for each of you.
What are the chances that another random person from the same broader gene pool shares one or two of the same 1024 couples to be 10th cousins? If you pick 1024 random couples and I pick 1024 random couples out of a population of, say, 1 million couples (a reasonable number for the number of actively-childbearing couples in the US about 10 generations ago), what are the chances we won't pick any of the same couples? This is an example of the
Multiple-type Collison Problem, which leads to a lot of ugly math.
(Aside:
This guy uses a modified estimate to show that there's a 1.4% chance of anyone in the world being 11th cousins, near certainty of any two people whose ancestors were among the half million Ashkenazi Jews being 10th cousins, and near certainty of anyone on earth being 16th cousins.)
The short version: two people whose ancestors were primarily from the US in the late 1700s are about 40% likely to be 10th cousins. So it's not surprising to find that Obama and Palin are 10th cousins; what's surprising is to find out who the link is between them.
