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Jandila comments on 2011 Less Wrong Census / Survey - Less Wrong
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Most people don't actually know their karyotype, and are often surprised to learn that it's not always what you assume. You can't necessarily infer chromosomes from external appearance and self-identification reliably; you have to look at the actual chromosomes to be sure.
If I'm not mistaken, you don't need a DNA test for this. A cell sample under a strong microscope will show the barr bodies for XX (this won't distinguish XXY, but that's pretty rare).
Looking at the barr bodies is not a karyotype test. A test that can't detect whether or not someone is not XX/XY sufficient to actually tell you the information you need to know your chromosome type.
Yes, in terms of strict probability most people will be one of those. The test of the method is how well it handles edge cases (not at all); this is of considerably greater importance when you're talking about those edge cases.