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ill_be_strong – Well said. And thank you for the compliments. However…although you are correct that person A may not have kept up with the average, and in fact there is pretty much a 50% chance that he did NOT, in order for the probability to continue decreasing as the generations progress, as you have shown may occur between gen 1 and gen 2, his children would also need to be significantly beaten by the average, and their children, and their children, etc. Statistics’ greatest friend is large numbers – because there have been so many generations since Yishmael lived, the overall expected probability has become more and more likely to occur; in all likelihood, about half of the generations had fewer than average number of children, and the other half had more (and a minute few were dead on with average). For this reason, it is EXTREMELY likely that every person of arab ethnicity, and perhaps every person in the world, is in one way or another a direct descendant of Yishmael.