Birthday buddies

For each integer \(n\geq 1\), let \(p(n)\) denote the probability that, in a room with \(n\) people chosen at random, two (or more) of them share a birthday. Find the value of \(n\) for which \(p(n)\) is as close as possible to \(\frac{1}{2}.\)

Details and assumptions

Every year has 365 days, a person's birthday is equally likely to fall on any of the possible 365 dates, and the birthdays of different people are independent of each other.

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