For a population to be in Hardy-Weinberg equilibrium, all of the following must be true:

1. No mutations - The DNA cannot change from one generation to the next.

2. No gene flow - There is no movement of individuals or alleles into or out of the population.

3. No genetic drift - The population must be large enough to avoid random changes in allele frequencies.

4. Random mating - Individuals must pair by chance, not according to traits.

5. No natural selection - All genotypes must have equal chances of surviving and reproducing.

Real populations don't meet these conditions, which is exactly is why the Hardy-Weinberg model is useful as a null hypothesis.

For a gene with two alleles (usually written as A and a), the frequencies of the alleles in a population can be represented as:

• p = frequency of the dominant allele (A)

• q = frequency of the recessive allele (a)

Since there are only two alleles,

p + q = 1

The expected genotype frequencies in the population can be calculated with:

p² + 2pq + q² = 1

Where:

• p² = frequency of individuals with genotype AA

• 2pq = frequency of individuals with genotype Aa

• q² = frequency of individuals with genotype aa

These equations allow you to calculate allele and genotype frequencies, and to test whether a population is evolving.