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(Please note that this is interpreted to be "ANY two people with ANY birthdate the same".) Yes, even if they "celebrate" their common birthday on some other day. In fact you only need 367. A random selection of people will have several pairs who share the same birthday, and several dates (especially the magic "February 29") are almost certain to be omitted. Statistical probability requires a very large sample, so that if 365,000 people were polled, you would have close to 1000 of each separate date, and as many as 250 with February 29th. (a 1-in-1461 average).

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Q: In a room filled with 370 people will there be at least two people who celebrate their birthdays on the same day?
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