The Ramanujan Journal

, Volume 28, Issue 2, pp 223–238

A (probably) exact solution to the Birthday Problem

Article

DOI: 10.1007/s11139-011-9343-9

Cite this article as:
Brink, D. Ramanujan J (2012) 28: 223. doi:10.1007/s11139-011-9343-9

Abstract

Given a year with n≥1 days, the Birthday Problem asks for the minimal number https://static-content.springer.com/image/art%3A10.1007%2Fs11139-011-9343-9/MediaObjects/11139_2011_9343_IEq1_HTML.gif such that in a class of https://static-content.springer.com/image/art%3A10.1007%2Fs11139-011-9343-9/MediaObjects/11139_2011_9343_IEq2_HTML.gif students, the probability of finding two students with the same birthday is at least 50 percent. We derive heuristically an exact formula for https://static-content.springer.com/image/art%3A10.1007%2Fs11139-011-9343-9/MediaObjects/11139_2011_9343_IEq3_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1007%2Fs11139-011-9343-9/MediaObjects/11139_2011_9343_IEq4_HTML.gif.

Keywords

Birthday ProblemRamanujan’s Q-functionAsymptotic analysis

Mathematics Subject Classification (2000)

41A6060C0565D10

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

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