Squibs
I don't own anything having to do with Harry Potter, and no money is being made from this fanfiction. (I am also not a geneticist or professional mathematician-although algebra tutoring is available upon request…)
So why are squibs so rare? The short answer is they're really not, at least comparatively. I am in the midst of writing an actual story, The Red Haired Boy, in which a much younger Arthur and Molly Weasley are dealing with the implications of what to do with a squib child. It builds upon Paterfamilias by Vera Rozalsky and The Other Draco by Swallow B., with the implication that we only think that squibs are rare because they are hushed up, hidden away, given up for adoption, or otherwise disposed of. However, I am a hopeless geek and have been unable to resist the siren call of the actual mathematics of the subject (with a little science thrown in for good measure). Since these do not exactly lend themselves to flowing narrative, I decided to post a separate essay on the subject. Read at your own risk!
Ron tells us that squibs are the inverse of muggleborns, but they are really rare. However, that is actually not true. Muggleborns are a significant portion of the wizarding population; the usual estimate seems to be about 25%. However they are an infinitesimal percentage of the general muggle population. Assume a total wizarding population of 20,000 (a very generous estimate-but I'd like to go with the idea that the population is at least marginally viable which Rowling's "about 3000" really is not). If a quarter of that group were muggleborn, those 5000 first-generation witches and wizards would make up approximately .008% of the 60 million residents of the United Kingdom (as of 2001). That's less than 1 in 10,000. If squibs were that rare, it is highly unlikely that we would have seen three in canon, even in passing. (In fact, statistically there would be 1.6 squibs for the entire wizarding population of Britain.) Even if we were to accept the idea that the three squibs mentioned in the stories were the only ones alive at the time (not likely) they would still be occurring at nearly twice the rate of muggleborn witches and wizards.
There is also the question of the actual genetics of magical inheritance. Rowling says that magical ability is a dominant trait, but if it were that simple it should have spread much further into the general population (before the split in 1692). Most essays I have seen on this subject try to force the (fictional) ability to do magic into Mendel's basic green pea/yellow pea model we all learned back in high school. However, in reality, very few traits are controlled by a single gene. (Just search "eye color genetics" on the web, and you'll find a much better explanation of polygenic traits than I could ever give.)
A more realistic, though still highly oversimplified, model would involve a combination of genes. For example, imagine magical ability to be governed by three separate genes, of which two must be dominant for the ability to manifest. A single gene (X) passed down from both parents has three possible combinations of dominant and recessive alleles, XX, Xx and xx. Two genes have nine possible combinations, XXYY, XXYy, XXyy, etc. Three have twenty- seven combinations, twenty-two of which would meet our imaginary criteria of at least two dominant alleles.
It is possible for a recessive or unstable gene to become the norm in an isolated population. That is why certain ethnic groups are prone to blond hair and blue eyes, or rare genetic diseases. The small size of the wizarding population, combined with the pureblood tendency toward inbreeding could easily lead to the "loss" of a necessary gene. Muggleborns, on the other hand would be likely to have a full complement of healthy genes. In that case, "pureblood supremacy" would eventually eliminate magical ability instead of strengthening it.
