For this question we are going to need to make use of the Hardy-Weinberg equilibrium equations. The equation we need to use is:

These numbers represent the percentages of each genotype found in a given population. We were given the values of  and  in the question.

After plugging the numbers into the equation, we can find the value of . This value will give us the frequency of homozygous dominant individuals.

We can solve this question using the Hardy-Weinberg equations:

In the second equation, corresponds to the frequency of homozygous dominant individuals, corresponds to the heterozygous frequency, and corresponds to the frequency of homozygous recessive individuals. We are given enough information to find each of these values from the question.

We can find the values of and by taking the square root of their squares.

We can solve this question using the Hardy-Weinberg equations:

is equal to the recessive allele frequency, while in the second Hardy-Weinberg equation corresponds to the frequency of the recessive phenotype.

The question tells us the number of dominant red snails and the number of recessive white snails. Using these values, we can find the frequency of the recessive phenotype.

From here, take the square root to find the value of .

Most of the information in the question is actually superfluous because we are given the final dominant allele frequency. The dominant allele frequency corresponds to the variable in the Hardy-Weinberg equations.

The question tells us that the dominant allele frequency after introduction of the predator is . Use this value in the first Hardy-Weinberg equation to solve for the recessive allele frequency, .

For a population in Hardy-Weinberg equilibrium, every trait follows the equations:

In these formulas,  represents the frequency of the dominant allele and  represents the frequency of the recessive allele.  represents the frequency of the homozygous dominant genotype,  represents the frequency of the heterozygous genotype, and  represents the frequency of the homozygous recessive genotype.

In this case, the individuals with blue eyes would be represented by the homozygous recessive genotype. Using this data, we can solve for the frequency of the recessive allele.

Use the frequency of the recessive allele to find the frequency of the dominant allele, .

It is impossible to determine the allele frequency from the given information.

The problem only tells the number of black beetles, but does not give any information that would allow us to find the total number of beetles in the population. We do not have the homozygous recessive population or the distribution of heterozygotes and homozygous dominant beetles. Given information on the number of white beetles would allow us to calculate the recessive allele frequency, and subsequently the dominant allele frequency.

Random mating is one of the five Hardy-Weinberg assumptions that help maintain equilibrium. If random mating occurs, in tandem with the other assumptions, we can reasonably assume that there will not be a shift in allele frequencies or distributions.

The other Hardy-Weinberg assumptions are that natural selection does not occur, mutation does not occur, genetic drift (gene flow) does not occur, and that the population size is large.

One of the five main assumptions is that mutations are negligible. This makes sense because, if a population is in Hardy-Weinberg equilibrium, evolution is not occurring. A low rate of mutations would help keep a population in equilibrium.

The five assumptions of Hardy-Weinberg equilibrium are a large population size, no natural selection, no mutation rate, no genetic drift, and random mating.

Using the Hardy-Weinberg equilibrium equations, you can determine the answer.

The value of gives us the frequency of the dominant allele, while the value of gives us the frequency of the recessive allele. The second equation corresponds to genotypes. is the homozygous dominant frequency, is the heterozygous frequency, and is the homozygous recessive frequency.

16% of the population shows the recessive phenotype, and therefore must carry the homozygous recessive genotype. We can use this information to solve for the recessive allele frequency.

We can use the value of and the first Hardy-Weinberg equation to solve for .

Knowing both  and , you can use the second equation to find the percent of heterozygous organisms in the population.

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:

We are told that 194 plants in a population of 200 demonstrate the dominant phenotype. This means that the sum of the homozygous dominant and heterozygous individuals is represented by the 194 plants described.

So, 97% of the plants show the dominant phenotype. This means that 3% must show the recessive phenotype.

Using this, we can find the frequency of the recessive allele, , and subsequently the dominant allele, .