The Hardy-Weinberg principle is a fundamental concept in population genetics that allows us to calculate genotype frequencies from known allele frequencies. The key equations involved in this principle are:

1. **Genotype Frequency of Homozygous Dominant (AA)**: \( p^2 \) 2. **Genotype Frequency of Heterozygotes (Aa)**: \( 2pq \) 3. **Genotype Frequency of Homozygous Recessive (aa)**: \( q^2 \)

In this context, \( p \) represents the frequency of the dominant allele (A), and \( q \) represents the frequency of the recessive allele (a). The relationship between \( p \) and \( q \) is defined by the equation \( p + q = 1 \).

For example, if the frequency of the A allele is 0.2 (thus \( p = 0.2 \)) and the frequency of the a allele is 0.8 (thus \( q = 0.8 \)), we can calculate the genotype frequencies as follows:

1. **Homozygous Dominant (AA)**: \[ p^2 = (0.2)^2 = 0.04 \]

2. **Heterozygotes (Aa)**: \[ 2pq = 2 \times 0.2 \times 0.8 = 0.32 \]

3. **Homozygous Recessive (aa)**: \[ q^2 = (0.8)^2 = 0.64 \]

After calculating these frequencies, we find:

- Frequency of AA: 0.04 - Frequency of Aa: 0.32 - Frequency of aa: 0.64

To ensure the calculations are correct, we can verify that the sum of all genotype frequencies equals 1:

\( 0.04 + 0.32 + 0.64 = 1.00 \)

This confirms that the calculations are accurate, and the genotype frequencies derived from the allele frequencies are consistent with the Hardy-Weinberg equilibrium. Understanding these calculations is essential for analyzing genetic variation within populations and predicting how allele frequencies may change over time.