Let’s add the peoples one by one. For each person you add, you add a small % of chance of getting a double birthday.
So if you enter a room of 22 peoples, one of those two are true:
1. There was already a double birthday
2. There was no double birthday, and you have 20/365 = 6% of having the same birthday than someone else there.
And sure, that 6% is not much, **but that 6% is “just for you”**. There was already 22 peoples in the room, all of them with their own probability of having the same birthday as someone else.
And sure, the computation is not as easy as “23 times 6%” because probabilities are more difficult than that to compute (there are correlations and stuff).
But the idea is the same: if everyone on the room has a small probability of something happening, and that those are truly “different events” (and not like “the probability of dying from a nuclear war” where everyone dies or live at once), the room has a big probability of that thing happening.
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