Genotype frequenciesp² + 2pq + q² = 1

Allele frequencies at a two-allele locusp + q = 1

What each term stands forp² = frequency of AA, 2pq = frequency of Aa, q² = frequency of aa

The usual way into a problem, from the recessive phenotypeq = √q², then p = 1 - q

A baseline for a population that is not evolving

Hardy-Weinberg equilibrium describes a stable, non-evolving state in which allele frequencies stay the same from generation to generation. It holds only when five conditions are met: no mutations, no gene flow, random mating, no genetic drift and no selection. Hardy and Weinberg themselves recognized that no natural population is immune to evolution, so the point of the equation is not to describe reality. It is to give a mathematical baseline against which a real population can be compared. If the observed frequencies deviate from the predicted ones, something on that list of five is not holding, and the population is evolving.

A worked example

In a plant species, violet flower color (V) is dominant to white (v). A field holds 1,000 plants and 160 of them have white flowers. Assuming the population is in Hardy-Weinberg equilibrium, find p and q, and the expected number of plants of each genotype.

  1. Start from the only genotype you can read directly. White is the recessive phenotype, so every white plant is vv and q² = 160 / 1,000 = 0.16.
  2. Take the square root for the recessive allele frequency: q = √0.16 = 0.4.
  3. Use p + q = 1 to get the dominant allele frequency: p = 1 - 0.4 = 0.6.
  4. Apply the genotype terms. p² = 0.6² = 0.36, so 0.36 × 1,000 = 360 plants are VV. 2pq = 2 × 0.6 × 0.4 = 0.48, so 480 plants are Vv. q² = 0.16, so 160 plants are vv, which matches the count the problem gave.
  5. Check both sums before stopping. 360 + 480 + 160 = 1,000, and 0.36 + 0.48 + 0.16 = 1.00, so the equation balances. Violet flowered plants number 360 + 480 = 840.

= p = 0.6 and q = 0.4, giving 360 VV, 480 Vv and 160 vv plants, which is 840 violet flowered and 160 white flowered.

Where students go wrong

The single most common error is treating q² as the allele frequency rather than the genotype frequency. The count you can read off a field or a class list is the recessive phenotype, and that count gives q², so the square root comes first and everything else follows. The second error is adding the wrong things: p + q = 1 applies to alleles, while p² + 2pq + q² = 1 applies to genotypes, and mixing them produces answers that look plausible and sum to nothing sensible. The third is forgetting the factor of 2 in 2pq, which quietly loses half the heterozygotes.

Where this leads next

This equation appears on the AP Biology equations and formulas sheet, alongside chi square, water potential and the population growth expressions, so it is usually met in the same breath as the chi square test that decides whether an observed deviation is large enough to take seriously. Conceptually it sits directly on top of Mendelian genetics: p and q are allele frequencies, and p², 2pq and q² are the same genotype proportions a Punnett square produces, scaled up from two parents to a whole population. The genotype and phenotype page covers why only q² is directly countable.

Common mistakes

  • Using the recessive phenotype count as q rather than q². Take the square root first; the count of recessive individuals gives q².
  • Dropping the 2 from 2pq, which halves the predicted number of heterozygotes.
  • Mixing the two equations, so that p + q is set against genotype frequencies or p² + 2pq + q² against allele frequencies.
  • Concluding a population is in equilibrium because the arithmetic works. The equation is a baseline; equilibrium also requires no mutation, no gene flow, random mating, no drift and no selection.

Related concepts

  • Dihybrid cross: A dihybrid cross follows two characteristics at once, starting from true-breeding parents that differ in both.
  • Punnett square: A Punnett square applies the rules of probability to predict the outcomes of a genetic cross.
  • Genotype vs phenotype: A genotype is an organism's underlying genetic makeup, including alleles that are not expressed.

Sources