"Very unlikely" is a severe understatement. With only 100k people, the odds of no one else having your birthday is 10^-120. With a million people, it is 10^-1192.
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?
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: