The Hardy-Weinberg principle gives a baseline prediction for genotype frequencies when allele frequencies are not being changed by evolutionary forces and mating is random with respect to the locus.

For two alleles $A$ and $a$ with frequencies

$$p+q=1,$$

random union of gametes gives

$$P(AA)=p^2,$$ $$P(Aa)=2pq,$$ $$P(aa)=q^2.$$

These terms come directly from the probability of combining gametes:

$$p^2+2pq+q^2=(p+q)^2=1.$$

Worked example

If

$$p=0.70,\qquad q=0.30,$$

then Hardy-Weinberg genotype frequencies are

$$P(AA)=0.49,$$ $$P(Aa)=2(0.70)(0.30)=0.42,$$ $$P(aa)=0.09.$$

In a population of 10,000 individuals, that corresponds approximately to 4,900 $AA$, 4,200 $Aa$ and 900 $aa$.

Under the idealized Hardy-Weinberg conditions, allele frequencies remain constant and these genotype proportions are regenerated after random mating.

The model assumes, for the locus being considered, a very large population, random mating, negligible mutation, negligible migration and no genotype-dependent differences in reproductive success.

Real populations need not satisfy all these assumptions. That is exactly why the model is useful: Hardy-Weinberg equilibrium is a null model. A substantial deviation can indicate that assumptions are violated, although the deviation alone does not identify which evolutionary process is responsible.