Mendelian Proportions in a Mixed Population
One line of algebra: left alone, a population's genes keep their balance.
A pure mathematician settled a biology argument with one line of algebra: left alone, a population's mix of genes simply stays put.
The big idea
Every gene comes in versions called alleles — say A and a. In a population, some fraction of all the copies are A and the rest are a; call those fractions p and q. Hardy showed that if individuals pair off at random, the proportions of the three genotypes — AA, Aa and aa — settle immediately into p², 2pq and q², and then never drift on their own. A common gene stays common; a rare one stays rare. Nothing about being "dominant" makes a version take over.
How it came about
In 1908 the geneticist Reginald Punnett was stumped by a question from the statistician Udny Yule: if a dominant trait is passed on, shouldn't it gradually spread until three people in four carry it? Punnett took the puzzle to his Cambridge cricket partner, the mathematician G. H. Hardy. Hardy saw the answer in minutes and dashed off a half-page letter to the journal Science, apologising for intruding on a subject he claimed to know nothing about. Unknown to him, a German doctor, Wilhelm Weinberg, had published the very same result earlier that year — which is why we now call it the Hardy–Weinberg law.
Why it mattered
Before Hardy, many feared that dominant traits would slowly swamp recessive ones out of existence — a worry that fed some ugly debates about heredity. His algebra showed there is no such force: inheritance by itself changes nothing. That made it the fixed reference point of population genetics. Whenever the genotype numbers in real data don't match p²:2pq:q², a biologist knows that some real force — natural selection, inbreeding, migration — must be at work, and goes looking for it.
A way to picture it
Imagine a huge bag of marbles, a fraction p of them blue (A) and the rest white (a). To make each new individual you reach in and grab two marbles at random. The chance of two blues is p × p, two whites q × q, and one of each 2 × p × q — that is all p²:2pq:q² is. Crucially, grabbing marbles doesn't repaint them: next time, the bag still holds the same fraction of blue. The gene pool's makeup is conserved no matter how the marbles happen to pair up.
Where it sits
Mendel (1866) had shown that traits come from discrete factors and that a single cross gives a 3:1 ratio; the danger was to mistake that ratio for a law about whole populations. Hardy and Weinberg corrected the leap. Their static equilibrium then became the launch pad for R. A. Fisher, J. B. S. Haldane and Sewall Wright, who in the 1920s and 30s added selection, mutation and chance to it and fused Mendel's genetics with Darwin's natural selection into the modern theory of evolution.
It is easy to see that the condition for this is q² = pr.
There is not the slightest foundation for the idea that a dominant character should show a tendency to spread over a whole population, or that a recessive should tend to die out.