My questin is a variation on the birthday paradox problem:
The difference being that here we want to know if two people have the same given birthday, not any same birthday. How would I solve this?
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1$\begingroup$ Well, it's easier to compute the probability that none of the $N$ are born on the chosen day, or that exactly one of the $N$ people were born on that day. $\endgroup$– luluCommented Oct 26, 2019 at 18:01
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1 Answer
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If you have n people, the possibility that no person is born on the 24th of december is $P_0=\frac{365}{366}^{n}$. The possibility that exactly one person is born on the 24th of december is $P_1=\frac{365}{366}^{n-1}\times \frac{1}{366}\times n$. Therefore the possibility that two or more people are born on the 24th of december is $P_{\geq2}=1-P_0-P_1$. From this you can solve for the minimum amount of people needed.
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$\begingroup$ Are you sure about the $n$ in $P_1$? $\endgroup$ Commented Oct 26, 2019 at 18:12
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$\begingroup$ For each individual, the probability that he is the only one born on the 24th is $\frac{365}{366}^{n-1}\times \frac{1}{366}$. There are n individuals, therefore we have to multiply this probability by n. $\endgroup$– T. SomCommented Oct 26, 2019 at 18:14