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TABLE 4.2 Hardy-Weinberg Proportions for a Locus with Three Alleles
Alleles  (and Frequencies)  in Eggs	Alleles (and Frequencies) in Sperm
	A1 (p1)	A2 (p2)	A3 (p3)
A1 (p1)	A1A1 (plpl)	A1A2 (p1p2)	A1A3 (p1p3)
A2 (p2)	A2A1 (p2p1)	A2A2 (p2P2)	A2A3 (p2p3)
A3 (p3)	A3A1 (p3p1)	A3A2 (p3p2)	A3A3 (p3p3)

of genotype D₂D₂ would be (0.109)² = 0.012, or 1.2%; as Table 4.1 shows, the observed fraction in this sample is 2.2%. The proportion of genotype D₂D₄ would be 2(0.109)(0.271) = 0.059, or 5.9%; the observed value is 4.6%. Neither of those differences is statistically significant. (Note that genotype D_1.3D_1.3 was not found in the black database of 224 persons. With multiple alleles and four or five loci, as with VNTRs, most genotypes are not found in any given database.)

The HW  relationship is easily stated symbolically. Using letter subscripts for generality, we let p_i and p_j be the population proportions of two alleles A_i and A_j. If capital letters designate the genotypic proportions, the HW expectations are

(4. 1a)
(4. 1b)

In words, the simple rule is: The proportion of persons with two copies of the same allele is the square of that allele's frequency, and the proportion of persons with two different alleles is twice the product of the two frequencies.

If for some reason a population does not exhibit HW proportions, as will be the case if mating in the previous generation(s) has not been random, only a single generation of random mating is needed to produce HW proportions. This is clear from Table 4.2, which shows that the proportions of gametes that unite to produce individuals in the next generation depend only on the allele frequencies, not the parental genotypes of the current generation. That property adds greatly to the usefulness of Equations 4.1, because it increases the probability that they are accurate. Populations from different parts of the world with different allele frequencies can be homogenized in a single generation, provided that mating is random. Of course, exactly random mating is very unlikely, but the equations are accurate enough for many practical purposes. In Chapter 5 we give estimates of the degree of uncertainty caused by departures from random mating proportions.

Table 4.1 shows how close actual populations come to HW proportions for DQA. The deviations from HW expectations are not great. In the white population, there is a small but statistically significant excess of homozygotes (P » .03);