In two-child families containing at least one boy, the expected probability that such a family has two boys is 1/3, provided that the boy/girl (B/G) ratio is 1.0 and the population to which they belong has a binomial distribution of BB, (BG + GB), and GG families. It is commonly known that in most human populations the sex ratio at birth (i.e., the ratio of the number of boys to the number of girls) is greater than 1.0. Teachers and textbook writers seldom discuss the more realistic expected distributions in populations where the sex ratio is greater than 1.0. We present data from two federal surveys with sex ratios greater than 1.0 and find that the observed proportions of two boys in families of size 2 with at least one boy range from 0.3335 to 0.3941. It has been reported in the literature that the probability (p) of a male birth is subject to both within-sibship variation (Poisson variation), for which our data are suggestive, and possibly also between-sibship variation (Lexis variation). These deviations (biases) from the assumptions of a simple binomial distribution are involved in the calculation of values of p and standard 95% confidence intervals, thereby foiling attempts to make reliable statistical inferences from the data. Analysis of the data is also complicated by family planning that falsifies the assumption of randomness in the binomial gender distribution model. Families of size 2 (and their sex composition) are often discussed in a wider context. Overpopulation in some parts of the world has caused mass starvation and threatens to do the same worldwide unless the birth rate drops to agriculturally sustainable levels. Even if every woman of fertile age has only two children on average from now on, the world’s population is predicted to continue growing toward 9 billion people by 2050. Other sociological problems are bound to follow. Although the birth rate in China has recently dropped, the average age of the population has risen, so that by 2035 it is projected that for each person over age 65 there will be just three working-age people. Furthermore, China’s one-child policy has already led to a sex imbalance where there is a large excess of men for whom marriage and parentage is denied.