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

A (probably) exact solution to the Birthday Problem

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Abstract

Given a year with n≥1 days, the Birthday Problem asks for the minimal number http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1007%2Fs11139-011-9343-9/MediaObjects/11139_2011_9343_IEq4_HTML.gif .