In 2006 there were a total of 4,265,996 births, 15-19 year olds gave birth to 435,427 babies. 40-44 year olds gave birth to 105,476 babies. By Bayes Theorem (using 1/1250 and 1/45 as the odds), given that we know that a child has down syndrome, the probability that the mother was 40-44 is 43% and that the mother was 15-19 is 6%.So, even taking into account the fact that a woman Bristol's age is more likely to be impregnated than someone Sarah's age is, women of Sarah's age range are more than six times as likely to have Down babies are girls Bristol's age are to have them.
Half Sigma's 'conspiracy theorizing' about Sarah Palin playing the birth of her grandson off as her own needs to address one crucial statistical tendency arguing strongly against it--babies with Down Syndrome are more than an order of magnitude more likely to be born to women in their forties than they are to be born to teenage girls. This table shows that one in forty births to a woman Sarah's age (44) will be Down babies. In contrast, only one in 1,250 births to girls daughter Bristol's age (18) will be. So Sarah is more than 30 times as likely to have had a baby with Down's Syndrome than Bristol is.
As HS' case is built on lots of suggestive circumstantial evidence, he should at least address the improbability (not the impossibility, of course) of Sarah Palin not being the mother of the Down baby.