I checked, and for 10,000 people, the probability is 99.99999996%. For the record, I tried 19,175; the result was so close my computer rounded it off to 100%.
I really can’t see the “some birthdays are more likely than others” making much difference. I’d be willing to bet that any differences would not have much effect on the 2,287 figure, and that that figure can safely be rounded off to 2,300 and be considered accurate (I really don’t think the differences could possibly push the actual amount anywhere close to 2,400).
If we also include leap day, that could make a difference. I don’t really want to bother with including that extra calculation from the beginning, but if we assume that 1 in 4*365 = 1460 people are born on leap day (a safe assumption, IMO), we get that if we have 2300 people gathered together, there’s about a 79.3% chance somebody was born on leap day.
So, in an attempt to completely kill this problem (including leap day part), I’m gonna make the following assumptions:
Birthdays are equally distributed over all days of the year, with the exception of leap day, which is the birthday of exactly 1/1460th of the population.
The fact that birthdays aren’t uniformly distributed doesn’t matter much, it’s close enough.
P(all 366 days are represented among n people) =
P(all 365 regular days are represented and leap day is represented) =
P(all 365 regular days are represented) * P(leap day is represented, given that all other 365 days are represented)=
f(n) * g(n),
where f(n) is the function I posted in my first post, and
g(n) = (1 - (1459/1460)^(n-365)).
This breaks 50% at 2473 people.
I’m confident the answer in the real world is in the ballpark of 2500.