Hardy-Weinberg is one line of algebra with a surprising amount of consequence packed into it. If two alleles are present at frequencies p and q, and mating is random, the genotypes settle at fixed proportions.
p² + 2pq + q² = 1
The affected proportion for a recessive condition is q², the homozygous recessive class. The carriers are 2pq, heterozygous and unaffected. Those two expressions behave completely differently as q gets small, and that is the whole point.
Squaring a small number
Take a recessive allele at a frequency of 0.01. Squaring it gives an affected rate of 0.010 per cent, or one in ten thousand. The Hardy-Weinberg calculator puts the carrier frequency at 1.98 per cent, which is about one person in fifty.
So for every person with the condition there are 198 who carry it and will never know. The ratio is 2p over q, which for any rare allele is close to 2 divided by q, and it grows as the condition gets rarer. At q = 0.02 the affected rate is 0.040 per cent and carriers are 3.92 per cent, a ratio of 98 to 1. Rarer conditions have proportionally more carriers, not fewer.
Why rare recessive conditions persist
This is the answer to a question that puzzles people about natural selection: if a recessive condition is severe, why has it not been eliminated. Because selection can only act on the q² who express it, while the overwhelming majority of copies of the allele are sitting in the 2pq who are unaffected and reproduce normally.
At q = 0.01 the arithmetic is stark. Every person carries two copies of the gene, so the allele accounts for 2q, or 0.02 copies per head of population. Of those, the heterozygotes hold 0.0198 and the affected hold 0.0002, which means 99 per cent of all copies of the allele are sitting in people who will never show the condition. Even complete selection against the affected class removes a hundredth of the allele pool each generation, which is why elimination takes an extremely long time and why the frequency is so stable in practice.
What the model assumes
Random mating, no selection, no mutation, no migration and an infinite population. None of those is exactly true anywhere, which is the point: Hardy-Weinberg is a null model. Its value is that a population which departs from it is telling you something, and the size and direction of the departure is the evidence.
Consanguinity is the departure that matters most for recessive conditions, because related parents are far more likely to share a rare allele than two people drawn at random. That raises the affected rate well above q² without changing the allele frequency at all, which is exactly the kind of thing the null model is there to make visible.
For working from observed genotype counts back to the allele frequency, there is allele frequency, and for the single-cross case rather than the population one, the monohybrid Punnett square gives the familiar 3:1 and the carrier probabilities within a family.