Hardy–Weinberg equilibrium with example

The Hardy–Weinberg principle states that both allele and genotype frequencies in a population remain in an equilibrium (are constant) as long as no specific disturbing influences are introduced.

Those disturbing influences could be mutations, non-random mating, selection, limited population size, gene flow and random genetic drift. Because these influences are universal in real life Hardy-Weinberg principle is never absolutely accurate. Although it sounds more theoretical it can help predict things with reasonable error.

Take for example a single gene that could occur as either dominant allele (A) or recessive(a) and their frequencies are denoted by p and q respectively. Assuming population equilibrium, frequency of occurrence of AA(A homozygosity) is p^2 and frequency of aa (a homozygocity)is q^2. Similarly the frequency of Aa (heterozygous) should be 2pq. 

To better understand this consider an autosomal recessive mutation that causes sickle cell anemia in homozygous recessive children. The parents of a boy with this mutation wants to know the probability of their grandchildren inheriting the disease. In order to determine the chance that the child will reproduce with a carrier of the recessive mutation we can use the above equation.  In order to know this we should know the incidence of heterozygous girls with this mutation and this can be derived if the incidence of homozygous (disease) state is known.

Let us assume that the homozygous state occurs at the rate of 64 per 10,000 people. Hence the occurrence of aa (disease) is 0.0064 hence q^2 0.0064 Hence q = 0.08

p+q=1 hence p=1-q=1-.08=.92. According to this, AA is p^2, which is .8464. Heterzygotic frequency is 2pq, which is 0.1472

Hence the chance that the young boy will mate with a heterozygous girl are about 14% and half of all their kids will be homozygous hence the chance that their kids will have the disease is about 30%