If you give a color badge to each person in the world and you have 365 colors to choose from, how accurately would a specific color identify a person?

What is 1-(1/365)?

Should probably involve the population there somewhere?

Sort of.

If I have a population of 1, or <= 365 and uniform dist of birthdays, the probability of a date being a valid identifier is 100%.

If I have a population of 365 with randomly distributed birthdays, then it is going to be the probability of not sharing a birthday with any of them (very low)

Once you can guarantee at least 2 people with each birthday the identifier is simply zero percent effective.

Wait, what about leap years?