Hardy-Weinberg Principle

The Hardy-Weinberg principle, formulated independently in the very same year, 1908, by the British mathematician G. H. Hardy and the German physician Wilhelm Weinberg, provides population genetics with a precise, quantitative mathematical description of the allele and genotype frequencies that would be expected within a population that is NOT, at that particular moment, undergoing any evolutionary change at all — that is, a hypothetical population within which the frequencies of the various alleles present remain exactly constant from one generation to the very next. This stable, unchanging condition is generally called Hardy-Weinberg equilibrium, or, equivalently, genetic equilibrium, and the Hardy-Weinberg principle supplies the essential mathematical benchmark against which the allele and genotype frequencies of any REAL population can subsequently be compared, in order to determine directly whether that real population is, in fact, currently undergoing evolutionary change or not.

Consider, for the sake of a concrete working example, a single gene possessing just two distinct alleles within a population: a dominant allele, conventionally denoted A, present at some frequency p within the overall population, and a recessive allele, conventionally denoted a, present at frequency q, where, by the very definition of these two frequencies as proportions of one single, complete gene pool, p + q = 1 must always hold true. If mating between individuals within this population is entirely random with respect to this particular gene — meaning, specifically, that an individual's own genotype at this gene has no systematic influence whatsoever on which particular mate that individual happens to pair with — then simple probability theory can be used to directly predict the expected frequency of each of the three possible genotypes among the resulting offspring. The probability that any given offspring happens to receive an A allele from BOTH of its two parents (thereby resulting in the homozygous genotype AA) is simply p × p = p², since the two parental contributions of alleles are independent events. Correspondingly, the probability that an offspring receives an a allele from both parents (resulting in the homozygous genotype aa) is q × q = q². Finally, the probability that an offspring receives one A allele and one a allele (resulting in the heterozygous genotype Aa) must be calculated slightly differently, because there are actually TWO distinct, equally likely ways in which this particular combined outcome can occur — either an A allele from the mother combined with an a allele from the father, OR, alternatively, an a allele from the mother combined with an A allele from the father — giving a combined heterozygote probability of (p × q) + (q × p) = 2pq.

Because every single individual within the population must, necessarily, fall into exactly one of these three mutually exclusive genotype categories (AA, Aa, or aa), and no individual can simultaneously belong to more than one category, these three separate genotype probabilities must together sum up to exactly 1 (that is, to 100% of the entire population), giving the now-famous Hardy-Weinberg equation:

This equation, it is worth noting explicitly, is nothing more mathematically exotic than the simple algebraic (binomial) expansion of the expression (p + q)²; and since p + q is already known to equal exactly 1 by the very definition of p and q as two complementary allele frequencies together accounting for an entire gene pool, it follows immediately and necessarily that (p+q)² = 1² = 1 as well, precisely confirming the equation above. …