Hardy-Weinberg (Genotype)

Core idea

Overview

The Hardy-Weinberg genotype equation describes the expected distribution of genotypes in a population that is not evolving. It mathematically demonstrates how allele frequencies (p and q) relate to the frequencies of homozygous dominant (p²), heterozygous (2pq), and homozygous recessive (q²) individuals.

When to use: Apply this equation when evaluating if a population is in genetic equilibrium or if evolutionary forces are at work. It assumes a large population, random mating, no mutation, no migration, and no natural selection.

Why it matters: It serves as a critical null hypothesis in evolutionary biology, allowing scientists to detect deviations that indicate evolution is occurring. This helps researchers identify which specific factors, like sexual selection or genetic drift, are influencing a species.

Symbols

Variables

p = Dom. Allele Freq, q = Rec. Allele Freq, $p^{2}$ = Homo Dom Freq, 2pq = Hetero Freq, $q^{2}$ = Homo Rec Freq

p

Dom. Allele Freq

Variable

q

Rec. Allele Freq

Variable

p^{2}

Homo Dom Freq

Variable

2pq

Hetero Freq

Variable

q^{2}

Homo Rec Freq

Variable

sum

Check Sum

Variable

Walkthrough

Derivation

Formula: Hardy-Weinberg Genotype Frequencies

Predicts genotype frequencies (p², 2pq, q²) from allele frequencies (p and q) when a population is in Hardy–Weinberg equilibrium.

The population is large and mating is random.
No mutation, migration, or selection affects the gene.
Allele frequencies p and q are constant across generations.

1

State the Allele Frequencies:

Let p be the frequency of the dominant allele and q be the frequency of the recessive allele; together they make 1 (100%).

p + q = 1

2

Square the Relationship for Diploid Genotypes:

Because individuals have two alleles per gene, genotype frequencies come from multiplying allele probabilities (squaring the binomial).

(p + q)^{2} = 1

3

Expand to Obtain Genotype Terms:

p² is homozygous dominant, 2pq is heterozygous, and q² is homozygous recessive.

p^{2} + 2 pq + q^{2} = 1

Result

p^{2} + 2 pq + q^{2} = 1

Source: OCR A-Level Biology A — Genetics, Evolution and Ecosystems

Free formulas

Rearrangements

Solve for $p$

Make p the subject

p = - q \pm 1

Combine the terms into a perfect square, take the square root, then subtract q to isolate p. The exact algebraic rearrangement has two branches.

Difficulty: 4/5

Solve for $q$

Make q the subject

q = - p \pm 1

Rewrite the left side as a perfect square, take the square root, then subtract p to isolate q. The exact algebraic rearrangement has two branches.

Difficulty: 4/5

Solve for $p^{2}$

Make P the subject

p^{2} = 1 - 2 pq - q^{2}

Subtract the heterozygous and homozygous recessive terms from both sides to isolate p squared.

Difficulty: 2/5

Solve for 2pq

Make H the subject

2 pq = 1 - p^{2} - q^{2}

Subtract the homozygous dominant and homozygous recessive terms from both sides to isolate 2pq.

Difficulty: 2/5

Solve for $q^{2}$

Make Q the subject

q^{2} = 1 - p^{2} - 2 pq

Subtract the homozygous dominant and heterozygous terms from both sides to isolate q squared.

Difficulty: 2/5

Solve for 1

Make sum the subject

1 = p^{2} + 2 pq + q^{2}

The total genotype frequency is already isolated on the right-hand side of the displayed formula.

Difficulty: 1/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

This view plots the heterozygote frequency 2pq with q = 1 - p, giving the classic downward-opening parabola. The peak occurs at p = 0.5, where genetic variation is highest because dominant and recessive alleles are equally common.

Graph type: parabolic

Why it behaves this way

Intuition

A statistical picture where the proportions of the three possible genotypes (homozygous dominant, heterozygous, homozygous recessive)

p

The frequency (proportion) of the dominant allele in the population's gene pool.

A higher 'p' indicates a greater prevalence of the dominant allele, increasing the likelihood of dominant traits appearing in the population.

q

The frequency (proportion) of the recessive allele in the population's gene pool.

A higher 'q' indicates a greater prevalence of the recessive allele, increasing the likelihood of homozygous recessive individuals.

p^{2}

The expected frequency of homozygous dominant individuals (e.g., AA genotype) in the population.

This term represents the probability of an individual inheriting two dominant alleles, one from each parent, assuming random mating (p ×p).

2pq

The expected frequency of heterozygous individuals (e.g., Aa genotype) in the population.

This term accounts for the two ways a heterozygote can form: inheriting a dominant allele from one parent and a recessive from the other (p ×q), or vice versa (q ×p).

q^{2}

The expected frequency of homozygous recessive individuals (e.g., aa genotype) in the population.

This term represents the probability of an individual inheriting two recessive alleles, one from each parent, assuming random mating (q ×q).

1

The total proportion of all possible genotypes in the population.

The sum of the frequencies of all possible genotypes (homozygous dominant, heterozygous, homozygous recessive) must equal 1, representing 100% of the population.

Free study cues

Insight

Canonical usage

The Hardy-Weinberg equation expresses relationships between allele and genotype frequencies, which are dimensionless proportions of a population.

Common confusion

Students sometimes confuse allele/genotype frequencies (proportions) with the actual count of alleles or individuals. The equation works with proportions, not raw counts.

Dimension note

All terms in the Hardy-Weinberg genotype equation represent frequencies or probabilities, which are inherently dimensionless quantities. They are proportions of a whole (the population).

Unit systems

$p$ dimensionless - Represents the frequency of the dominant allele in a population, expressed as a proportion (0 to 1).

$q$ dimensionless - Represents the frequency of the recessive allele in a population, expressed as a proportion (0 to 1).

$p^{2}$ dimensionless - Represents the frequency of the homozygous dominant genotype, expressed as a proportion (0 to 1).

2pqdimensionless - Represents the frequency of the heterozygous genotype, expressed as a proportion (0 to 1).

$q^{2}$ dimensionless - Represents the frequency of the homozygous recessive genotype, expressed as a proportion (0 to 1).

One free problem

Practice Problem

In a stable population where the frequency of the recessive allele (q) is 0.3, calculate the frequency of the homozygous dominant genotype (P).

Rec. Allele Freq0.3

Solve for: $P$

Hint: First determine the dominant allele frequency (p) using p + q = 1, then square it to find P.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When estimating genotype distribution from allele surveys, Hardy-Weinberg (Genotype) is used to calculate Check Sum (should be 1) from Dom. Allele Freq, Rec. Allele Freq, and Homo Dom Freq. The result matters because it helps compare populations or ecosystems and decide whether the system is growing, stable, or under stress.

Study smarter

Tips

Always calculate q (the recessive allele frequency) first when starting from phenotype data.
Remember that p + q must always equal 1 for a biallelic system.
The term 2pq represents the frequency of carriers in a population.
Check your work by ensuring p² + 2pq + q² equals exactly 1.

Avoid these traps

Common Mistakes

Forgetting the 2pq term.
Using percentages instead of decimals.
Not checking that p + q = 1 before calculating.
Applying to evolving populations (only valid when assumptions are met).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Predicts genotype frequencies (p², 2pq, q²) from allele frequencies (p and q) when a population is in Hardy–Weinberg equilibrium.

Apply this equation when evaluating if a population is in genetic equilibrium or if evolutionary forces are at work. It assumes a large population, random mating, no mutation, no migration, and no natural selection.

It serves as a critical null hypothesis in evolutionary biology, allowing scientists to detect deviations that indicate evolution is occurring. This helps researchers identify which specific factors, like sexual selection or genetic drift, are influencing a species.

Forgetting the 2pq term. Using percentages instead of decimals. Not checking that p + q = 1 before calculating. Applying to evolving populations (only valid when assumptions are met).

When estimating genotype distribution from allele surveys, Hardy-Weinberg (Genotype) is used to calculate Check Sum (should be 1) from Dom. Allele Freq, Rec. Allele Freq, and Homo Dom Freq. The result matters because it helps compare populations or ecosystems and decide whether the system is growing, stable, or under stress.

Always calculate q (the recessive allele frequency) first when starting from phenotype data. Remember that p + q must always equal 1 for a biallelic system. The term 2pq represents the frequency of carriers in a population. Check your work by ensuring p² + 2pq + q² equals exactly 1.

References

Sources

Wikipedia: Hardy-Weinberg principle
Campbell Biology, 12th Edition
Campbell Biology by Urry, Cain, Wasserman, Minorsky, and Reece
Hardy-Weinberg principle Wikipedia article
OCR A-Level Biology A — Genetics, Evolution and Ecosystems

Overview

Variables

Derivation

State the Allele Frequencies:

Square the Relationship for Diploid Genotypes:

Expand to Obtain Genotype Terms:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Allele Frequency

Frequently Asked Questions

Sources