Abstract
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 
.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmed, S.E., McIntosh, R.J.: An asymptotic approximation for the birthday problem. Crux Math. 26, 151–155 (2000)
Berndt, B.C., Rankin, R.A. (eds.): Ramanujan: Essays and Surveys. American Mathematical Society, Providence (2001)
Bernoulli, J.: Ars Conjectandi. Basel (1713). Translation: Wahrscheinlichkeitsrechnung. Verlag von Wilhelm Engelmann, Leipzig (1899)
Davenport, H.: The Higher Arithmetic, 8th edn. Cambridge University Press, Cambridge (2008)
de Bruijn, N.G.: Asymptotic Methods in Analysis, 3rd edn. Dover, New York (1970)
Dilcher, K.: Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials. J. Approx. Theory 49, 321–330 (1987)
Flajolet, P., Grabner, P.J., Kirschenhofer, P., Prodinger, H.: On Ramanujan’s Q-function. J. Comput. Appl. Math. 58, 103–116 (1995)
Gamow, G.: One, Two, Three… Infinity. Viking, New York (1947)
Graham, R., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading (1994)
Halmos, P.R.: I Want to Be a Mathematician. Springer, New York (1985)
Holmes, S.: Random Birthday Applet. Available electronically at http://www-stat.stanford.edu/~susan/surprise/Birthday.html
Katz, V.J.: A History of Mathematics. Harper Collins, New York (1993)
Knuth, D.E.: The Art of Computer Programming, vol. I, 2nd. edn. Addison-Wesley, Reading (1973)
Kubina, J.M., Wunderlich, M.C.: Extending Waring’s conjecture to 471,600,000. Math. Comp. 55, 815–820 (1990)
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (1974)
Littlewood, J.E.: A Mathematician’s Miscellany. Methuen, London (1953)
Mathis, F.H.: A generalized birthday problem. SIAM Review 33, 265–270 (1991)
Ramanujan, S.: Question 294. J. Indian Math. Soc. 3, 128 (1911)
Ramanujan, S.: On Question 294. J. Indian Math. Soc. 4, 151–152 (1912)
Singmaster, D.: Sources in Recreational Mathematics. Published electronically at www.g4g4.com/MyCD5/SOURCES/SOURCE4.DOC
Sloane, N.J.E.: On-Line Encyclopedia of Integer Sequences. Published electronically at http://oeis.org/
Tsaban, B.: Bernoulli numbers and the probability of a birthday surprise. Discrete Appl. Math. 127, 657–663 (2003)
van der Corput, J.G.: Diophantische Ungleichungen. I. Zur Gleichverteilung modulo Eins. Acta. Math. 56, 373–456 (1931)
von Mises, R.: Ueber Aufteilungs- und Besetzungs-Wahrscheinlichkeiten. Rev. Fac. Sci. Univ. Istanbul 4, 145–163 (1939)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brink, D. A (probably) exact solution to the Birthday Problem. Ramanujan J 28, 223–238 (2012). https://doi.org/10.1007/s11139-011-9343-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-011-9343-9