Probability is just a model in lieu of the actual data with many built-in assumptions. Probability is a shortcut finding, collecting or generating data. My point is, if you have the data or a very large faction thereof, use it. Don't guess using probabilities.

Right, but it's really not about birthdays at all.

It's specifically about coincidences in a randomly distributed set.

It just says "birthday" in the common name because that's a relatable, if imperfect, example.