Postprint version. Published in The American Statistician, Volume 59, Issue 2, May 1, 2005, pages 180-182. Copyright © 2005 American Statistical Association. The definitive version is available online at: http://dx.doi.org/10.1198/000313005X42813.
Many probability and genetics textbooks pose standard questions about eye color, birth defects, sexes of children, and so on. Solutions to these questions, specifically about sexes, generally make two assumptions: first, that a randomly selected embryo is equally likely to be male or female; second, that the sexes of successive children from the same parents are independent. In other words, probabilists (and some geneticists) treat sexes of children like flips of a fair coin: two possible outcomes, each equally likely, with outcomes independent from trial to trial. But are these assumptions realistic? Demographic data suggest that neither a balance of sexes nor true independence exist in nature. Yet most textbooks, both in genetics and probability theory, continue to use the binomial distribution as an acceptable approximation for solving genetics problems involving live-birth sex ratios in species where sex is determined by an XX versus XY chromosome mechanism.We look at a widely circulated article in Parade magazine regarding the gender distribution in human families with two children and analyze comparable data from federal sources to show that such families do not conform to any binomial distribution. The sequence of investigations we take here could be followed in an introductory or intermediate probability and statistics course.