= Hardy-Weinberg principle
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{title2=$p^2,2pq,q^2;\quad p+q=1$}
{wiki=Hardy–Weinberg_principle}
= Hardy-Weinberg equilibrium
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{synonym}
= HWE
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= Hardy-Weinberg proportions
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The <Hardy-Weinberg principle> describes the <genotype> proportions produced by independently combining two parental <gametes> from the same <allele> pool. For <alleles> $A,a$ with <allele frequencies> $p,q$, where $p+q=1$, the <Hardy-Weinberg proportions> are
$$
P(AA)=p^2,\qquad P(Aa)=2pq,\qquad P(aa)=q^2.
$$
The two ordered <heterozygote> outcomes contribute $pq+qp$. These proportions sum to $(p+q)^2=1$, and their implied $A$ <allele frequency> is $p^2+(2pq)/2=p$. Reshuffling copies therefore preserves their frequencies.
For stable <Hardy-Weinberg equilibrium> across generations, use a large population with <random mating>, equal <allele frequencies> in the parental pools, ordinary <Mendelian segregation>, and no <natural selection>, <mutation>, <gene flow> or <genetic drift>. Negligible <genetic drift> is an idealization of large population size. The <one-generation random-mating equilibrium> needs independent union of <gametes>; persistence additionally requires processes that change <allele frequencies> to be absent.
At $p=0.70,q=0.30$, the predicted <genotype> proportions are $0.49,0.42,0.09$. For 100 sampled individuals the <expected values> of the counts are $49,42,9$, rather than the $25,50,25$ of a specified $Aa\times Aa$ cross. The latter fixes both parental <genotypes>; <Hardy-Weinberg proportions> average over population mating.
\Image[/hardy-weinberg-proportions.png]
{title=Hardy-Weinberg genotype proportions as allele frequency varies, highlighting p = 0.70}
An observed population need not have these proportions. For example, $70\ AA$, no $Aa$, and $30\ aa$ among 100 individuals still gives $p=0.70$, but has no <heterozygotes>. <Random mating> predicts $0.42$ <heterozygosity> in the next generation under the stated model. A finite sample can also depart by chance, so a discrepancy is a starting point for investigating <inbreeding>, <population stratification> or <genotyping error>, rather than proof of one cause.
Hardy's 1908 note made the conceptual payoff explicit: a dominant <phenotype> does not by itself make its <allele> spread. <Complete dominance> describes expression; changes in frequency require a separate mechanism such as <natural selection>. See https://www-leland.stanford.edu/~huatang/gene244/readings/HardyWeinberg.pdf[Hardy, Mendelian Proportions in a Mixed Population, Science 28 (1908), 49–50].
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