Calculus meets the birthday problem

Abstract

What is the likelihood P(n, T) that at least two people in a gathering of n people are born within a given time T of each other? In particular, for T = 24 hours what is the smallest n for which P(n, T) is at least 50%? The well-known classic version of the birthday problem asks for the smallest number of people needed to give a better than 50% chance of at least one birthday match with the assumptions that the birthdays are independently selected from a discrete uniform distribution over 365 days. Using the calculus tool of the limit, we refine two characterisations for P(n, T) and show that they give consistent results with each other and with the classic birthday problem as well. At first thought P(n, 24 hours) should answer the classic birthday question. Yet consider this experiment: suppose Andrea was born at noon on June 1, then potential matching birth times (for the other n - 1 people) with her birth time extend from noon on May 31 to noon on June 2, a period of 48 hours! What this period suggests is that P(n, 24hours) exceeds the classic answer. But, by how much? Read on.