Thank you for the opportunity to help you with your question! the solution is as such:
In this equation (p² + 2pq + q² = 1), p is defined as the frequency of the dominant allele and q as the frequency of the recessive allele for a trait controlled by a pair of alleles (A and a). In other words, p equals all of the alleles in individuals who are homozygous dominant (AA) and half of the alleles in people who are heterozygous (Aa) for this trait in a population. In mathematical terms, this is
p = AA + ½Aa
Likewise, q equals all of the alleles in individuals who are homozygous recessive (aa) and the other half of the alleles in people who are heterozygous (Aa).
q = aa + ½Aa
Because there are only two alleles in this case, the frequency of one plus the frequency of the other must equal 100%, which is to say
p + q = 1
Since this is logically true, then the following must also be correct:
p = 1 - q
There were only a few short steps from this knowledge for Hardy and Weinberg to realize that the chances of all possible combinations of alleles occurring randomly is
(p + q)² = 1
or more simply
p² + 2pq + q² = 1
In this equation, p² is the predicted frequency of homozygous dominant (AA) people in a population, 2pq is the predicted frequency of heterozygous (Aa) people, and q² is the predicted frequency of homozygous recessive (aa) ones.
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