Wow, that's an excellent question. Not much is written about this, possibly because race and health care can be controversial and emotionally charged issues. I'm not an epidemiologist (someone who studies the causes and characteristics of human diseases), but here's my understanding of the situation.

I don't think that minority children are more prone to Down's syndrome because, as you say, the chromosomal abnormality that causes the syndrome is random. So I think a sort of "mis-observation" may be going on. However, a number of factors contribute to what kids you might encounter who have Down's syndrome, and some of those factors might correlate very strongly with race or other factors.

For example, some racial groups might be more likely than others to test for Down's syndrome during pregnancy and abort fetuses that test positive for Down's. Someone's religion or her access to adequate health care might affect her decision to abort a Down's fetus. Once a Down's baby is born, her parents' economic situation might decide how fully she is integrated into society--if she's educated and well cared for at home, you might run across her around town; if she ends up in some sort of institution, however, you might never encounter her. Also, Down's children from better economic backgrounds (thus enjoying better health care) probably live longer, so you're more likely to encounter them than babies who die young.

So, there might be some correlations with race, but not direct correlations between race and chromosomal abnormalities. More likely, the correlations are between Down's and access to health care or between Down's and economic prosperity. Unfortunately, however, Caucasians on average enjoy better health care and greater economic prosperity than most minorities do. That's probably at the root of your observation.

I hope that helps. Keep up with the great questions!