but doesn't 1:1666 mean the success rate is more than 99%?
Asking about the posession of an XX chromosome should separate the people almost 50:50, the rest is something else.
On the other hand, maybe there are just <1% people who don't identify them self with being male or female, so the questions would make no difference...
Not XX and not XY: one in 1,666 births
Klinefelter XXY: one in 1,000 births
There are other relevant cases, not necessarily measured here, like mosaicisms and chimerism.
"Do you self-identify as female?" is a more answerable question.