This is all very interesting, but it requires knowing the allele frequencies in a population. The power of the Hardy-Weinberg equilibrium formulas is in their ability to allow us to determine these frequencies by simple observation of the population.
When we see an organism with the dominant phenotype, we do not know whether we are looking at a homozygote (BB) or a heterozygote (Bb). When we see one with the recessive phenotype, though, we know it has the genotype bb. If we determine the proportion of individuals in the population with the bb genotype, it is a simple step to calculate the frequency of b. Let's work backward through our example above. If we determine that 1 percent of the population is homozygous recessive, bb = 0.01.
Since the frequency of bb organisms is the product b X b, we can take the square root of this number to get the frequency of b:
Since B + b must equal 1 (assuming there are only two alleles), B must be 0.9. With these calculations in hand, we can predict what the frequency for each of the other genotypes should be: BB is 0.81, and Bb is 0.18.
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