They don’t go off last names
The sequence your DNA, literally decoding it into all the base pairs that make up your DNA. They look in the database for people with similar DNA. You get half your DNA from your mom and half from you dad, so if they find someone who has ~50% overlap with you, then it’s your mom or dad or a sibling. ~25% would be a aunt, uncle, niece or nephew, or grandparents. 12.5% would be a first cousin etc
https://kidshealth.org/en/parents/genetics.html and https://www.pbs.org/newshour/health/23andme-need-know should be good starts.
Sequencing the whole genome isn’t necessary because so much of it will match from just being human. They just need to check certain portions of the DNA, like seeing how a word is spelled on a book.
They’re making statistical estimations based on others’ profiles. If your set of variations matches others who report their ancestry coming from a certain part of the world, then it scores you as more likely coming from there too. If you match someone ~50% they might be a sibling, etc. And different amounts of match you can guess sharing a common ancestor however many generations back.
It requires both peoples DNA, they wouldn’t be able to tell you if your related to Elvis, unless some of Elvis’ kin has taken the test. Or lets say they would have no clue if your related to King George III as he is far back enough he never took the test, and his kin prolly hasn’t so it would be far down the line, at which point you’d have to ask the person your related to if they have relation to KGIII or hope their family tree is accurate.
Last names obviously don’t work as I’m not related to a president, but, have the same last name as a prior one. If you ask my family the best they say is, if you go up the tree really high, and then back down the tree, than maybe. Chances are My and the presidents wouldn’t match enough to give a good statistic.
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