The Hardy-Weinberg equations can be used to determine the expected frequency of genes and genotypes within a population, provided that only Mendelian segregation and recombination of alleles are at work. This is calculated mathematically using the equations:

Let's start with what we know and how it relates to these equations. We are told that the dominant allele frequency () is 87%, and then asked to find the frequency of homozygous recessive individuals ().

Known:

Unknown:

Use the second Hardy-Weinberg equation to solve for .

Square this value for find the frequency of homozygous individuals.

The Hardy-Weinberg equations can be used to determine the expected frequency of genes and genotypes within a population, provided that only Mendelian segregation and recombination of alleles are at work. This is calculated mathematically using the equations:

In this example, the expected frequency to be solved for is the heterozygote G^MG^J that is represented by the  component of the equation. We are told the frequency of the G^M allele in the population, allowing us to solve for the frequency of the G^J allele. It is not necessary to know which allele is dominant and which is recessive in this particular question since we are dealing only with genotypes (phenotype is irrelevant).

Using these values, we can calculate the value of .

There are multiple ways to solve this problem, but the easiest is to use the Hardy-Weinberg equations:

We are told the frequency of white beetles in the population. Using this value, we can find the recessive allele frequency. is equivalent to the homozygous recessive genotype frequency.

Use this value to solve for the dominant allele frequency.

We can use the Hardy-Weinberg equations to solve this problem.

We know that spiral shells are dominant, and that 36% of the population is homozygous for the spiral allele. This tells us that 36% of the population is homozygous dominant. The  term corresponds to the homozygous dominant percentage.

 is the dominant allele frequency. Now we can solve for , the recessive allele frequency.

The term  will give us the frequency of heterozygotes.

48% of the population is heterozygous.

Using the Hardy-Weinberg law to solve for allele frequency in populations, you can solve for the answer using the following two equations.

p is the fractional frequency of the dominant allele, q is the fractional frequency of the recessive allele, and q² is the fraction of the population that is homozygous recessive. q² is given in the question to be 0.04 (or 4%).

Random mating is a Hardy-Weinberg assumption, not nonrandom. The remaining answer choices are all standard assumptions for Hardy-Weinberg equilibrium. Nonrandom mating would allow certain alleles to be preferentially passed down over generations. 

The Hardy-Weinberg principle is a theory that describes how allele frequencies may change within a population absent of evolutionary mechanisms. The theorem is based on certain assumptions regarding the population in question. These assumptions include random mating, large population size, sexual reproduction, and the absence of migration, mutation and selection. It is exceedingly rare for all the Hardy-Weinberg assumptions to be met in nature, but this theory is a tool used to study allele frequencies within populations. 

In the Hardy-Weinberg theory, the two equations used are:

Here,  refers to the frequency of the dominant allele and  refers to the frequency of the recessive allele. Subsequently,  refers to the frequency of the homozygous dominant genotype,  refers to the frequency of the homozygous recessive genotype, and  refers to the frequency of the heterozygous genotype.

In Hardy-Weinberg equilibrium, deviations are violations of the assumptions of the Hardy-Weinberg theory. The assumptions of the Hardy-Weinberg theoru include random mating, large population size, sexual reproduction, and the absence of migration, mutation and selection. Therefore, deviations from the Hardy-Weinberg equilibrium from the options are mutation and migration.

The genetic make up of populations can be measured over time. While genetic changes only occur at the individual level, it is only populations that evolve since genetic changes take many generations to develop into phenotypic changes that may be observed as changing allele frequencies over time.