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In this table, the total frequency is just the frequency of each parental mating pair taken from the parental mating frequency table. We now need to partition this total frequency of each parental mating into the frequencies of the three progeny genotypes produced. Let's look at an example. Parents with AA and Aa genotypes will produce progeny with two genotypes: half AA and half Aa (you can use a Punnett square to show this is true). Therefore, the AA x Aa parental matings, which have a total frequency of 2XY under random mating, are expected to produce (x/2)2XY = XY of each of AA and Aa progeny. The same logic applies to all of the other parental matings. Notice that each row in the offspring genotype frequency table sums to the total frequency of each parental mating.

The columns in the offspring genotype frequency table are the basis of the final step. The sum of each column gives the total frequencies of each progeny genotype expected in generation t + 1. Let's take the sum of each column, again expressed in the currency of genotype frequencies, and then simplify the algebra to see whether Hardy and Weinberg were correct.

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