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This was a question asked by one of my readers. In a word, Yes.
Albinism is an independently arising mutation which renders the body incapable of producing pigment. It is passed on as a recessive gene, so a black parent can have an unpigmented child. In fact, is is the usual state of things that an albino child will have parents who are not themselves albinos.
Now, there is more to the question than this. What if a white woman gives birth to a white child--is there any chance that the father was black? The answer to that question is also Yes, but with qualifications.
A child carries genes from both of his parents, which in turn were inherited from the grandparents. A child with one white and one black grandfather, and one white and one black grandmother, can be anything from almost completely white to almost completely black. So, if a white woman gives birth to a white baby, there are three possible reasons (granting, along with the questioner, that the father himself is not an albino):
1) The father was purely white (most common reason).
2) The father was partly white (next common reason).
3) The father and mother carry the albinism gene. If this is the case, the child will have some of its father's physical traits, but without his skin colour.
There is another question to answer: can an albino mother have a black baby? Yes, because the gene for expressing skin colour is dominant. Only if the father also has the recessive gene for albinism, and only if he passes it on to his offspring, will the offspring be albino--unless a mutation arises anew in that generation lacking the genetic capability of producing skin colour.
UPDATE JULY 11, 2011 (and later as needed)
I'm getting a lot of hits on this post, so I'll go into even further detail.
1) Q: What causes a baby to be born albino?
A: There are two basic reasons why a baby could be born albino:
1. The most common reason: Both his mother and his father carry the gene for albinism, and both of them passed that gene on to him. Such parents only have a 25% chance of having an albino child, a 50% chance of having a child who carries the albino gene, and a 25% chance of having a child without the gene for albinism. Such are the statistical chances; I've not seen any studies that show the numbers as actually observed. The odds drop to 1 in 16 for 2 albino children in a row, and 1 in 64 for three; still, better odds than having twins, so it does happen.
2. The original reason: The first albino, way back in the mists of time, was the result of a mutation in his father's line that lost the ability to produce skin pigment, combined with a mutation in his mother's line that did the same thing. This did not need to happen in the same generation; the father could have been born with the defective gene and the mother's egg spontaneously lost its function, resulting in the first fully albino genotype. Or vice versa. This obviously happened in the past, so it could happen again, but because the human genome is designed to detect and correct errors before they are passed on, the chances are minuscule.
2) Q: Do albinos always have albino children?
A: Yes, with qualifications. The children of two albinos will always come out albino. Children of an albino and an albinism carrier have a 50 per cent chance of being albino themselves. A couple in which both carry the albino gene may have all albino children, but that possibility becomes less and less likely the more children they have. If a person without the albino gene has children with an albino, all their children will carry the albino gene, but none will be albinos, except in the case of #2 above.
I know of families who had several albino children, so it can happen. But I don't know of any albinos who married each other--although one would expect that to eventually happen [BUT: see link at the bottom of the page for examples]. It's possible to raise up a whole race of albinos--such as the Albino deer herd on an army base in New York. Or the albino rabbit. Such only happens, though, under special circumstances that make up for the survival advantages that all albinos lack.
3) Can someone be almost albino?
Not exactly; there are various degrees of albinism, but only three possible genetic states:
1. Both genes (one from the father, one from the mother) carry usable information on producing pigmentation (it's actually a bit more complicated, in that there are about a dozen genes in all that have influence on skin colour). The dominant gene will be evidenced in the person's skin colour.
2. One gene contains usable information on producing pigmentation, whilst the other is an albino gene, unable to produce pigment. This person will still have skin colour based on the dominant gene, but will be able to pass on the defective gene. There's no way of telling by looking at a person if he carries the albino gene or not, as it is unexpressed.
3. Both genes (one from each parent) are albino genes, and the person is unable to manufacture any pigment. He has ruddy white skin, blond hair, and blue eyes (the default colours when no pigment is present). All he can pass on to his children are albino genes, so all of his children will be either #2 or #3, depending on the other parent.
Now, some people have genes that are just barely able to produce pigment, so they will resemble an albino in many ways: ruddy white skin, blond hair, blue eyes. But they are capable of producing some melanin, so they will be able to tan slightly (although they will burn very easily, and are likely to freckle rather than tan). But these people are not technically albinos and are usually not referred to as such, except in jest.
4) Can white people be albino?
Albinism is the inability to manufacture melanin, the main pigment in skin, hair, and the iris of the eye. Regardless of what other genetic information may be present for producing a particular colour of skin, hair, or eye, the albino is unable to use it. Thus albinos are all the same in this respect, regardless of whether they would otherwise be considered "black" or "white." This is also why someone with only one albino gene is totally normal in appearance; the gene from the other parent is able to facilitate the production of however much melanin is called for by the rest of the genetic code.
5) If white is the presence of all colors, why do albinos not have pigment?
Pigments are defined by the colour(s) of light they reflect. A green pigment, for example, reflects yellow and blue light equally, absorbing all the rest. A black pigment absorbs all colours of light; the absence of any pigment reflects all colours of light.
6) Can albino children have freckles?
This is a tricky one, but the answer is "yes." The gene for freckles, which is dominant, overrides the inability to produce melanin in albinos that have it. These freckles can be larger and more numerous than they would be on others. But again, since the gene is dominant, there is no guarantee that just because one or both parents have freckles, that any given one of their children will. Only if one or both of the parents passes on a freckle gene will the child have them.
SUMMARY:
The changes that a child will be albino are generally 0%, 25%, 50%, and 100% as follows:
0% if one of the parents does not carry an albinism gene.
25% if both parents are carriers, but not albino
50% if one parent is albino and the other a carrier
100% if both parents are albino.
UPDATE FEB 2013:
I realise I left one matter hanging: why are newborn babies so light skinned, even if they are born to heavily pigmented parents? It has nothing to do with being born albino; normally babies don't form melanin in their skin for a while after being born. This, by the way, is also why all light-skinned babies start out with blue eyes, whether or not they have darker eyes later on.
UPDATE JUNE 2013
Inasmuch as there are actually four different genes, defects in which result in albinism, there are various combinations and degrees of albinism that I don't go into here. Above is just a general overview.
UPDATE MAY 2015
Inasmuch as this has been, since the day I published it, my most popular post, I've gone back and edited it a bit in the interests of accuracy.
There is another post on the topic
here.