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Suppose a room contains 𝑛 people. What is the probability that at least two people share the same birthday?

So this is actually a question on probability, but I'm really confused in the logic and wording of the solution, not the math part.

Let A be the event that at least two people share the same birthday. The way to solve this problem is actually to find the complement of A and then solve the equation 1 - Probability of the Complement of A.

The solution to this problem says that the complement of A is "the event that no one shares the same birthday (everyone has different birthdays)." From my understanding, a complement is equivalent to the negation or opposite of something. If the opposite of "at least" is "at most", why is the complement of "at least two people share the same birthday" equal to "no one shares the same birthday" and not "at most two people share the same birthday?"

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    Hi, welcome to Philosophy SE. This is more of a question for Math SE. The complement of "at least" (i.e. greater than or equal to) is not "at most" (less than or equal to), but "strictly less than". Strictly less than two people sharing a birthday means one or less people "sharing", i.e. no one sharing, since "sharing" with yourself or nobody is not sharing at all. – Conifold Oct 3 '19 at 8:36
  • Search “chances of two people sharing the same birthday”. Many answers appear in the results. – Mark Andrews Oct 3 '19 at 22:16
  • Thank you @Conifold, your explanation cleared things up – John Oct 6 '19 at 12:22
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The complement of "at least two" is not "at most two" but "at most one" (in general the complement of "at least N" is "at most N-1"), because "at least two" and "at most two" could be true at the same time. But to say that "at most one person shares the same birthday with themself" is to say that no two different people do. So it is correct to say that the complement of "at least two people ..." is "no two people ...".

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    This is correct, but Confold's phrasing of 'strictly less than two' as the opposite of 'at least two' in the comments is better. It extends to the case of the real numbers, and most other domains. – user9166 Oct 3 '19 at 21:45

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