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You've probably heard of the Birthday Paradox, that among 23 people the probability that two share a birthday is more than 50%.

This is easy to prove by calculating the complementary that no two of the 23 share a birthday. There are 365 options for the first birthday, then only 364 for the second, 363 for the third, etc. So the probability that all 23 people have different birthdays is (365×364×...×343)/365²³ ≈ 0.4927.

That calculation assumes the uniform probability distribution on birthdays, that is, that every day is equally likely to be a birthday. That's not a very realistic assumption for the actual distribution of people's birthdays, so it's worth thinking about what happens for other distributions. My instinct was that any non-uniform distribution has some higher probability clump, where birthdays would tend to bunch up with even fewer people; so that any non-uniform distribution is even "more paradoxical" than the uniform one.

This turned out to be easy to prove.

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#math #probability