, Volume 28, Issue 2, pp 223-238
Date: 18 Apr 2012

A (probably) exact solution to the Birthday Problem

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Given a year with n≥1 days, the Birthday Problem asks for the minimal number such that in a class of students, the probability of finding two students with the same birthday is at least 50 percent. We derive heuristically an exact formula for and argue that the probability that a counter-example to this formula exists is less than one in 45 billion. We then give a new derivation of the asymptotic expansion of Ramanujan’s Q-function and note its curious resemblance to the formula for .