Diploid organisms have two alleles of each gene, one from each parent, in their somatic cells. Therefore, each individual contributes two alleles to the gene pool of the population. The gene pool of a population is the sum of every allele of all genes within that population and has some degree of variation. Genetic variation is typically expressed as a relative frequency, which is the percentage of the total population that has a given allele, genotype or phenotype.

In the early 20^{th} century, scientists wondered why the frequency of some rarely-observed dominant traits did not increase in randomly-mating populations with each generation. For example, why does the dominant polydactyly trait (*E*, extra fingers and/or toes) not become more common than the usual number of digits (*e*) in many animal species? In 1908, this phenomenon of unchanged genetic variation across generations was independently demonstrated by a German physician, Wilhelm Weinberg, and a British Mathematician, G. H. Hardy. The principle later became known as Hardy-Weinberg equilibrium.

The Hardy-Weinberg equation (*p*^{2} + 2*pq* + *q*^{2} = 1) elegantly relates allele frequencies to genotype frequencies. For instance, in a population with polydactyly cases, the gene pool contains *E* and *e* alleles with relative frequencies of *p* and *q*, respectively. Since the relative frequency of an allele is a proportion of the total population, *p* and *q* add up to 1 (*p* + *q* = 1).

The genotype of individuals in this population is either *EE*, *Ee*, or *ee*. Hence, the proportion of individuals with the *EE* genotype is *p* × *p*, or *p*^{2}, and the proportion of individuals with the *ee *genotype is *q* × *q*, or *q*^{2}. The proportion of heterozygotes (*Ee*) is 2*pq *(*p* × *q* and *q* × *p*) since there are two possible crosses that produce the heterozygous genotype (i.e., the dominant allele can come from either parent). Similar to allele frequencies, genotype frequencies also add up to 1; therefore, *p*^{2} + 2*pq* + *q*^{2} = 1, which is known as the Hardy-Weinberg equation.

Hardy-Weinberg equilibrium states that, under certain conditions, allele frequencies in a population will remain constant over time. Such populations meet five conditions: infinite population size, random mating of individuals, and an absence of genetic mutations, natural selection, and gene flow. Since evolution can simply be defined as the change in allele frequencies in a gene pool, a population that fits Hardy-Weinberg criteria does not evolve. Most natural populations violate at least one of these assumptions and therefore are seldom in equilibrium. Nevertheless, the Hardy-Weinberg principle is a useful starting point or null model for the study of evolution, and can also be applied to population genetics studies to determine genetic associations and detect genotyping errors.

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