system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: trying to find a simple answer the birthday paradox and how having 23 people in a room means a 50% chance of two people sharing a birthday.
gpt
"share a birthday" is not a property that a person has, it's a property that a pair of people have. With 23 people in a room there are 253 pairs of people (maths calls this 23 choose 2). There's only 366 days that could be shared by each of those pairs, so it's quite unlikely that they're all distinct.

Those pairs of people aren't quite independent, so it's not so straight forward as to say there's a 253/366 chance of a shared birthday, but that view should make it a lot more intuitive that these are comparable numbers and a 50% chance is not insanity.