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When there’s 23 people in a group, the chances that 2 or more people have the same birthday is: $1-\bigl(\frac{364}{365}\bigr)^{253}\approx0.5005$ which is found by taking the chances that a pair doesn’t share the same birthday, and multiplying by itself 253 times (the 253 represents the 253 unique comparisons of birthdays), and then subtracting that value from 1 to get the chances that 2 or more people do share the same birthday. This approach came from here: https://betterexplained.com/articles/understanding-the-birthday-paradox/

I have been looking for different ways to go about finding the probability that 2 or more people in a group of 23 people have the same birthday.

I decided to think about the situation as 23 people each picking a number between 1 and 365, and finding the chances that 2 or more people pick the same number. So I figured I could calculate the probability that nobody picks the same number by doing: $$\prod_{n=1}^{23}\biggl(\frac{366-n}{365}\biggr)$$ Which equals the chances that none of of the 23 people pick the same number. Then I could do $$1-\prod_{n=1}^{23}\biggl(\frac{366-n}{365}\biggr)\approx0.5073$$ to get the probability that there is at least 1 match somewhere.

But why aren’t these values the same? Is the way I interpreted the original problem completely different than how it started? Or did I do some math incorrectly?

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    $\begingroup$ Where does the first formula come from? The usual expression, given $n$ people, is $1-n!\times \binom {365}n\big / 365^{23} $ which yields $0.507297234$ when $n=23$. $\endgroup$
    – lulu
    Commented Apr 27, 2020 at 0:25
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    $\begingroup$ The 253 pairwise comparisons are not independent, so the first calculation is incorrect. $\endgroup$
    – Brian Moehring
    Commented Apr 27, 2020 at 0:30
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    $\begingroup$ @lulu I do not believe that OP is aware that the first attempt was incorrect and is only good for an approximation rather than a true result. It looks like the OP believes both approaches to be valid ways to arrive at the true result. $\endgroup$
    – JMoravitz
    Commented Apr 27, 2020 at 0:31
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    $\begingroup$ @ Cotton, if you need justification for why the first approach is wrong... consider an extreme case of having $400$ people. By pigeon-hole principle we know that the probability of at least two people sharing a birthday will be exactly $1$. The approach used in the first answer however gives an answer not equal to $1$. It is near $1$, but not exactly equal to $1$ which we know should have been the answer given by any actually correct formula. Looking at what went wrong for this extreme case can give you insight as to what went wrong in the general case. $\endgroup$
    – JMoravitz
    Commented Apr 27, 2020 at 0:35
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    $\begingroup$ I will point out, in your link... they address the fact that this approach is actually incorrect. Read ahead to appendix A where they say "Appendix A: Repeated Multiplication Explanation (Geeky Math Alert!) Remember how we assumed birthdays are independent? Well, they aren’t." $\endgroup$
    – JMoravitz
    Commented Apr 27, 2020 at 0:39

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Your first approach is dead wrong. Consider the case of 3 people. If the first two don't share a birthday, they have 2 different birthdays. If the third person doesn't share a birthday with the first person, it is slightly more likely than normal that the third person will share a birthday with the second person.

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