system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: How does the birthday paradox work?

The birthday paradox “vaguely” states that the odds of having the same birthday in a room of 20 people is about 50%. I tried finding the answer alone but no explanation I read helped.

Edit: Thank you everyone for your great expectations. I think I get the gist of it now.
gpt
So. Let's talk about probability for a second. We're trying to calculate the probability that at least two people in a room share a birthday.

At first glance, it looks like a problem that's hard to even state in words, let alone compute.

But one of the fun things about probability is that you can solve a complex probability question P(A) by instead solving its *complement,* P(A') -- its opposite.

In this case, our P(A), what we want to find out, is the probability that at least two people in the room share a birthday.

Its opposite is "The probability that *nobody in the room* shares a birthday." This is our P(A'), and it's *much* easier to solve -- either nobody shares a birthday, or at least two people do.

For two people, the probability that they do not share a birthday is given by:

    P(A') = 365/365 * 364/365

This is because the second person can have any birthday that isn't the first person's.

Adding a third person gives us

    P(A') = 365/365 * 364/365 * 363/365

and so on. It turns out that, after adding the 23^rd person, P(A') becomes about 49.27%. Because this is the opposite of what we want to solve, subtract it from 1:

    P(A) = 1 - P(A')
    P(A) = 1 - .4927, ∴
    P(A) = .5073

From there, we conclude that if you have 23 people in the room, the odds that at least two of them share a birthday is just over 50%.