I am aware of the birthday paradox (more than 50% likeliness of two people sharing the same day of birth). My question is somehow related to this, but with more variables (and unknowns).
Given a total population of lets say 50 million people, I am interested in people who share the same first and last name as well as the same date of birth. I presume that the likeliness of such a combination to exist in reality is close to 100% given the large population.
However, what I would like to estimate (no precision required here), is how many of those cases exist.
I guess it strongly depends on the distribution of names and date of birth. So lets assume that the most frequent first name has a 1.5% occurence and possibly can be best described using an inverse sigmoid function (not sure about the exact distribution). Roughly the same distribution applies to the last name.
How could I possibly calculate such an estimate and what else would I need to know?