TITLE: The birthday paradox
QUESTION [7 upvotes]: I would like a better understanding of the famous birthday paradox. 
"What is the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday?"
I understood the first part, where the probability reaches 100% when the number of people reaches 367 by the pigeonhole principle.
But I am not understanding the explanation beyond that.
How do they say that the probability is 99.9% with 70 people and 50% with 23 people? And how do you further generalize the answer?
And why is it a "paradox"?

REPLY [3 votes]: The paradox is that the rate of growth doesn't match our common sense. We expect that the way to count the number of possibilities for people to have the same birthday is directly from the number of people. However, in reality it's based on the number of pairs of people, which grows much faster than the number of people.
Banach Tarski shows you the derivation which really is that the numerator is the number of people pairs, but still over 365.