Scarlet tiger moth (Callimorpha dominula). Photo credit: copyright Shane Farrell, England. 
Population geneticists are interested in the processes driving evolution, notably natural selection and genetic drift. Evolution is defined as the change in allele frequencies within a population over time. In natural populations allele frequencies may remain roughly constant, or they may fluctuate (sometimes dramatically) due to selection, drift, mutation, etc. In the absence of these forces, the frequency of a given allele within a population will remain constant after one generation of random mating. The frequency of such allele can be determined using the HardyWeinberg Law, which was independently derived by British mathematician Godfrey Hardy and German physician Wilhelm Weinberg in 1908. The HardyWeinberg equilibrium has been applied to animal populations (such as the scarlet tiger moth, right), and is also used in genetic counseling. 
Consider a gene locus in a diploid population with two possible alleles, A and
a. Let p and q represent the frequency of the A and a alleles in the population,
respectively. As there are only two possible alleles, p and q, we know that
p + q = 1.
The three genotypes possible are AA, Aa, and aa. The HardyWeinberg Law states that after one generation of random mating, the frequencies
of the three genotypes in the population are given by:
f_{AA} = p^{2},
f_{Aa} = 2pq,
f_{aa} = q^{2}.
An individual has genotype AA if both parents contribute an A allele. Likewise, if both parents contribute an a allele then the individual has genotype aa. This can happen one way, and the probability that an individual is
homozygous for allele A is the frequency of the A allele in the population
squared, and the probability that an individual is
homozygous for allele a is the frequency of the a allele in the population
squared.
An individual has genotype Aa if either an A allele is inherited maternally
and an a allele is inherited paternally, or if an a allele is inherited maternally
and an A allele is inherited paternally. The probability of having an A and a
allele is pq (the product of their frequencies in the population), and since a heterozygous
genotype can arise two different ways, the frequency of genotype Aa is
2pq. Notice that because p+q = 1, we also know that (p+q)^{2} = p^{2} +2pq +q^{2} = 1.
A note of caution
The HardyWeinberg Law assumes that the population mates randomly, is of infinite size, and that there is no mutation, no migration, and no natural selection. 
Use the information above to answer the following questions:
Determine the frequencies of the aa genotype after one generation of random mating.
Determine the frequency of the Aa genotype after one generation of random mating.
Find the frequency of the A allele that will maximize the frequency of the Aa genotype.
Find the frequency of the A and a alleles in a given population.
Find the frequency of the A allele that will maximize the mean fitness of a population.
Find the frequency of the A allele that corresponds to a given mean fitness.
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