The Strong Birthday Problem Revisited

TL;DR

This work revisits the Strong Birthday Problem, seeking the minimum group size $n$ (for $m$ possible days) such that every person shares a birthday with someone with probability at least $1/2$. It develops a combinatorial formula via inclusion-exclusion, a three-term recurrence for counting distributions with collision and singleton days, and a fruitful connection to associated Stirling numbers ${n race k}_{\ge r}$ to derive alternative recurrences. The authors implement these methods with dynamic programming and validate numerical results against direct combinatorial calculations, providing efficient computation for moderate and large $n$. The results have practical relevance for problems in fingerprint matching and queueing, and the work suggests extensions including fast approximations and nonuniform birthday models for future research.

Abstract

We revisit the Strong Birthday Problem (SBP) introduced in [1]. The problem is stated as follows: what is the minimum number of people we have to choose so that everyone has a shared birthday with probability at least 1/2? We derive recurrence relations to compute the probability, and further show a nice connection to the associated Stirling numbers of the second kind to derive additional recurrences. We implement the recurrences using dynamic programming as well as compute the values using the combinatorial formula, and provide numerical results.