The Strong Birthday Problem Revisited

TL;DR

This paper revisits the Strong Birthday Problem, developing new computational methods and recurrence relations to efficiently analyze the probability that every individual shares a birthday with at least one other in a group.

Contribution

It introduces novel recurrence relations linked to Stirling numbers, improving computational efficiency for the Strong Birthday Problem analysis.

Findings

New recurrence relations derived for the problem

Efficient dynamic programming implementation demonstrated

Numerical evaluations show scalability and practical efficiency

Abstract

We revisit the Strong Birthday Problem (SBP) introduced by DasGupta'05, which asks for the minimum population n required such that, with a probability of at least 1/2, every individual in the group shares a birthday with at least one other person. Formally, we develop and analyze computational frameworks to determine the probability that in a group of n people with birthdays distributed over m days, each day either has two or more birthdays or is birthday-free. We derive both counting-based and probability-based recurrence relations to solve this problem and establish a novel connection to associated Stirling numbers of the second kind. This relationship is exploited to derive new, more efficient recurrences. Finally, we implement these recurrences using dynamic programming, provide analysis of their asymptotic complexities, and present numerical evaluations that demonstrate the practical efficiency and scalability of our proposed approaches.