Reminds me of the research that shows that 87% of people in the US can be uniquely identified with only three pieces of information: date of birth, gender, and zip code [1].
[1]: https://dataprivacylab.org/projects/identifiability/paper1.p...
ZIP codes contain maybe 40K residents [0] (many contain fewer) and there have been around 25K days in the last 70 years. Sure births are not evenly distributed, but still...
[0] https://www.unitedstateszipcodes.org/images/comparison-of-po...
I think you're making the assumption that all three data points are needed for all 87%. But obviously some people can be uniquely identified based on just {zip, date or birth}, such that gender isn't necessary.
So the distribution could e.g. be 8% same, 8% opposite, 5% both, 79% neither, and explain the original numbers without triggering the paradox.
87% of the time, there are no others on my birthdate or there is one other and opposite gender.
13% of the time is 1 same gender or more of either or both.
To clarify: your date of birth includes the year. It’s more specific than your birthday, which we usually think of as just day & month.
In other words, even if one person is extraordinarily tricky to find [0], their share of the total un-findable-ness does not diffuse outwards to help anybody else.