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An allele is one of the alternative inherited states at a genetic locus, often distinguished by a difference in DNA sequence. A population can contain many alleles, although a two-copy diploid individual carries at most two at that genetic locus, one inherited from each parent. See NHGRI: allele.

Count copies, not just individuals, when computing allele frequency: in 100 individuals with genotype counts $49AA$, $42Aa$, $9aa$, there are $2(49)+42=140$ copies of $A$ among 200, so $p_A=0.70$. The heterozygotes contribute to both allele counts. This converts observed genotypes into the population quantity needed by the Hardy-Weinberg principle.

Combine the untransmitted maternal and paternal alleles into a hypothetical control genotype, paired with the child's transmitted genotype. Even when two heterozygous parents have a heterozygous child and parent-specific phase is ambiguous, the combined untransmitted genotype is unambiguously heterozygous. Such pseudo-controls are matched family controls, not independent random individuals; Hardy-Weinberg principle is therefore not guaranteed merely by this construction.

Genetics studies how genes are inherited, how their alleles vary, and how that variation contributes to phenotypes. Its central distinction is between an inherited description, the genotype, and what an organism actually expresses, the phenotype. Following alleles through a pedigree answers a family-level question; following allele frequencies through generations leads to population genetics and evolution.

The beauty of Mendelian segregation is that discrete inherited copies can generate predictable statistical patterns without making every offspring identical. A single genetic locus already connects meiosis, probability and observable variation. The Hardy-Weinberg principle then supplies a population baseline: it separates reshuffling inherited copies from processes such as natural selection that change their frequencies. These simple models are useful precisely because their assumptions can be stated and checked.

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 areThe 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.

An observed population need not have these proportions. For example, $70AA$, no $Aa$, and $30aa$ 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 Hardy, Mendelian Proportions in a Mixed Population, Science 28 (1908), 49–50.

The two parental chromosomes are independent, with probabilities $1-\theta$ and $\theta$ for alleles $A$ and $B$. Therefore the Hardy-Weinberg principle gives