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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.

For a specified characteristic, complete dominance means that the heterozygote has the same phenotype as one of the two homozygotes. It does not mean that the dominant allele is more common, more advantageous, or transmitted more often. Penetrance specifies the conditional probability of an expressed phenotype given a genotype and is a separate modeling assumption.

A genotype records the inherited alleles carried by an individual at one or more genetic loci. At a two-copy diploid genetic locus with alleles $A,a$, the possible unordered genotypes are $AA,Aa,aa$. The first and last are homozygotes; the middle is a heterozygote. The notation records inherited states, without yet asserting their phenotypes. See NHGRI: genotype.

An unordered $Aa$ combines the two parental orders $Aa$ and $aA$. This small bookkeeping point explains the factor two in Hardy-Weinberg proportions and in an $Aa\times Aa$ cross. If parental origin or arrangement across several genetic loci matters, a phase-known genotype retains extra information about the two haplotypes.

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.

Heterozygosity means that the two allele states at a diploid genetic locus differ; an individual with that property is a heterozygote. Heterozygous parental genotypes provide alternative alleles for a transmission disequilibrium test.

A heterozygote has greater genotype fitness than either corresponding homozygote. In the deterministic random-mating two-allele viability-selection model this gives an attracting interior allele frequency, unlike the repelling interior frequency produced by underdominance.

A homozygote carries two identical alleles at a locus; a heterozygote carries different alleles. Homozygosity of an observed haplotype is identity by state, and need not be identity by descent.

Genetic recombination erodes linkage disequilibrium by reshuffling alleles between haplotypes, while preserving marginal allele frequencies in the idealized model. Let $f_{AB,t}$ denote the haplotype frequency in the gamete pool of generation $t$, let $p,q$ denote its marginal allele frequencies, and let $r\in [0,1/2]$ be the constant recombination fraction between two autosomal genetic loci. Assume random mating within one population, nonoverlapping generations, fair Mendelian segregation, equal reciprocal gamete production, the same $r$ in both sexes, and a deterministic population large enough to neglect genetic drift. Exclude natural selection, mutation and gene flow. ThenHere $p$ and $q$ remain constant: recombining changes which alleles occur together, not their total numbers. The other haplotype frequencies change by $+r{D}_t$ for $Ab,aB$ and $-r{D}_t$ for $ab$. Consequently the joint probability distribution stays within its allowed bounds and approaches the product of its marginal distributions when $r>0$.

For the original positive example $p=0.60,q=0.40,D_0=0.12$, a recombination fraction $r=0.10$ gives $D_1=0.108$ and $D_{10}\approx 0.04184$. The negative example $D_0=-0.12$ has exactly the opposite sequence: genetic recombination reduces the magnitude without reversing the sign. With $r=0$, the initial linkage disequilibrium persists in this model; even independent assortment at $r=1/2$ only halves $D$ each generation. A population association does not vanish after one generation merely because the genetic loci are unlinked.

The geometric sequence gives a generation scale: for $r>0$, the continuous interpolation reaches $|D_0|/2$ at $t_{1/2}=\log (1/2)/\log (1-r)$. Actual generations are integers; the first at or below half is $\lceil t_{1/2}\rceil$. For small $r$, $t_{1/2}\approx (\log 2)/r$. Because the denominator of allelic correlation stays fixed, $r_{\mathrm{LD},t}^2=(1-r{)}^{2t}{r}_{\mathrm{LD},0}^2$; its decay is faster than the magnitude of $D_t$.

A short derivation uses random mating to assign frequency $2f_{AB}{f}_{ab}$ to the double heterozygote $AB/ab$ and $2f_{Ab}{f}_{aB}$ to $Ab/aB$. Relative to $r=0$, each loses or gains an $AB$ gamete fraction $r/2$, respectively. Their net contribution is $r(f_{Ab}{f}_{aB}-f_{AB}{f}_{ab})=-rD$. The other genotypes produce no net change in $f_{AB}$, proving the recurrence and, by iteration, its solution.

This temporal model does not identify elapsed generations from physical separation alone. The recombination fraction depends on genetic recombination, including recombination hotspots, while population stratification, natural selection, gene flow or genetic drift can create or maintain linkage disequilibrium. Repeated sampling across time would also require accounting for sample size and haplotype estimation. A cross-sectional plot of association against distance between genetic loci is therefore a different observation from the idealized trajectory plotted here.

In ordinary diploid meiosis, the two copies at a genetic locus separate: each gamete receives one copy. With fair transmission, a heterozygote $Aa$ produces gametes carrying $A$ or $a$ with probability $1/2$ each. This is a transmission mechanism conditional on a parental genotype, rather than a statement about how common either allele is in a population.

For the one-locus cross $Aa\times Aa$, assume independent parental transmissions, no new mutation, and no differences in fertilization or survival between the resulting genotypes. Multiplying the two gamete probabilities gives four equally likely ordered outcomes: $AA,Aa,aA,aa$, each with probability $1/4$. Combining the two orders of the heterozygote givesOnly under the additional ideal model of complete dominance of $A$, complete penetrance, and no environmental changes to the scored characteristic do $AA$ and $Aa$ share the dominant phenotype. Its probability is then $0.25+0.50=0.75$. The inheritance calculation alone does not determine a phenotype or predict a generic disease risk.

These are probabilities, not a requirement that every four offspring contain one $AA$, two $Aa$ and one $aa$. For 100 independent offspring under the ideal model, the expected values of the three counts are $25,50,25$; the $aa$ count has a binomial distribution with standard deviation $\sqrt{100(0.25)(0.75)}\approx 4.33$. Each new transmission keeps the same conditional probabilities given the parental genotypes.

In his paper published in 1866, Mendel reported 5,474 round and 1,850 wrinkled pea seeds: $5474/7324\approx 0.7474$ round, close to the ideal $0.75$. Following descendants also distinguished two inherited classes within the round phenotype, illustrating why genotype cannot always be read from appearance.^[ref] At several genetic loci, transmission patterns additionally depend on genetic linkage and genetic recombination.

Underdominance means that the heterozygote has lower genotype fitness than both corresponding homozygotes. In a deterministic randomly mating biallelic population, it produces an unstable interior allele-frequency threshold separating the two fixation outcomes, unlike heterozygote advantage.

With homozygote fitness advantages $a,b>0$ over the heterozygote, the allele frequency $u$ of the first allele has an unstable threshold $b/(a+b)$. Below it that allele decreases; above it it increases. In a discrete viability-selection generation the same numerator gives $u^{\prime }-u$ after division by mean genotype fitness. Equal homozygote advantages give $\dot{u}=su(1-u)(2u-1)$, with stable endpoints and unstable midpoint.