The birthday paradox works because you are looking for a pair that has the same birthday, regardless of what date that birthday is. It doesn't work in the habitable planet sense because you would be essentially locking the birthday to a fixed date.
And with a billion people... apparently, there are over a million zeroes between the decimal place and the first non-zero number. That is a pretty damn small probability.
Here is the code I used in python 3:
from decimal import Decimal
print((Decimal(364) / Decimal(365)) ** num_people)There's other things as play. Imagine you, born on July 15th, walked into a room with 1000 people. You'd be fairly confident someone will be born on July 15?
What if I then told you it was the annual astrology get together of Capricorns?
It’s exactly what you would expect from classical combinatorics with cards with or without replacement. Your term “confident” is vague.
I produced a series called Thinking Mathematically on youtube that makes all of this and other related topics clear for anyone... I recommend checking it out!
https://m.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA
And here are the notes for it
http://magarshak.com/math/numbers.pdf
http://magarshak.com/math/sets.pdf
http://magarshak.com/math/logic.pdf
# if we exclude all people born leap years
Why would we assume a uniform distribution of birthdays? For example, birthdays occurring on the 31st of a month are probably less likely to occur on average given that every month does not have 31 days. This is just one example and doesn't even go into seasonality of conception cycles.
1 - ((365 - 1)/365)^253 = 0.5005
Not sure how that looks for the worlds population probably averages out pretty well regardless.
But if you aren't certain about that, it makes it less likely that everyone else with your birthday has been ritually murdered or otherwise systematically excluded from the room, and slightly more likely that you're at a convention dedicated to people with your birthday.