The Ramanujan Journal

, Volume 28, Issue 2, pp 223–238 | Cite as

A (probably) exact solution to the Birthday Problem

  • David BrinkEmail author


Given a year with n≥1 days, the Birthday Problem asks for the minimal number Open image in new window such that in a class of Open image in new window students, the probability of finding two students with the same birthday is at least 50 percent. We derive heuristically an exact formula for Open image in new window 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 Open image in new window .


Birthday Problem Ramanujan’s Q-function Asymptotic analysis 

Mathematics Subject Classification (2000)

41A60 60C05 65D10 


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity College DublinBelfield, Dublin 4Ireland
  2. 2.Copenhagen Business CollegeFrederiksbergDenmark

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