The Birthday Problem and the Birthday Person Problem: Intuition, Probability and Cryptographic Applications
Abstract
The present work discusses the birthday problem and the birthday person problem, two classical problems in probability theory widely used to illustrate situations in which human intuition diverges from mathematical results. Taking the work of [1] as its main reference, the study analyzes the probability of birthday coincidences within groups of people, considering a year with 365 days and assuming that all birth dates are equally likely. The birthday problem seeks to determine the probability that at least two people in a group share the same birthday, showing that, even in relatively small groups, the probability of a coincidence increases rapidly due to the large number of possible pairwise comparisons among individuals. In contrast, the birthday person problem investigates the probability that a previously selected person shares the same birthday as another member of the group. This probability increases more slowly, since the number of relevant comparisons is restricted to the pairs formed by the selected person and the remaining members of the group. Despite the interpretative differences, both problems share a similar mathematical foundation based on combinatorial analysis and probability theory. In addition to the theoretical development, numerical results and graphical representations are presented to illustrate the behavior of the probabilities involved and to visualize how the likelihood of coincidences increases as the number of people grows. Finally, applications related to computing and cryptography are discussed, particularly the study of collisions in hash functions and the so-called birthday attack. The discussion highlights the importance of probabilistic reasoning in understanding apparently unlikely phenomena, while also emphasizing how the growth in the number of possible combinations directly influences the results obtained.
Related
- A Novel Hyperchaotic System Derived from Asymmetric Bidirectional Coupling of Two Lorenz Oscillators: Synchronization Analysis and Cryptographic Applications
- Matrix concentration inequalities and free probability II: Two-sided bounds and applications
- A Walk From Free Probability to Matrix Discrepancy II: Weaver's Problem and the Kadison-Singer Conjecture
- High-Probability Nash Regret for Decentralized Learning in Markov α-Potential Games: Episodic and Fully Online Asynchronous Algorithms with Applications to Markov Congestion Games
- Birthday Estimators Are Uniformly Optimal for Renyi Entropy
BibTeX
@misc{silva2026birthday,
title = {{The Birthday Problem and the Birthday Person Problem: Intuition, Probability and Cryptographic Applications}},
author = {Alisson Garcia da Silva and Iarythssa Duarte de Araújo and Francisco Márcio Barboza},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38749},
eprint = {2610.38749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38749},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Birthday Problem and the Birthday Person Problem: Intuition, Probability and Cryptographic Applications
AU - Alisson Garcia da Silva
AU - Iarythssa Duarte de Araújo
AU - Francisco Márcio Barboza
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38749
UR - https://arxiv.org/abs/2610.38749
AB - The present work discusses the birthday problem and the birthday person problem, two classical problems in probability theory widely used to illustrate situations in which human intuition diverges from mathematical results. Taking the work of [1] as its main reference, the study analyzes the probability of birthday coincidences within groups of people, considering a year with 365 days and assuming that all birth dates are equally likely. The birthday problem seeks to determine the probability that at least two people in a group share the same birthday, showing that, even in relatively small groups, the probability of a coincidence increases rapidly due to the large number of possible pairwise comparisons among individuals. In contrast, the birthday person problem investigates the probability that a previously selected person shares the same birthday as another member of the group. This probability increases more slowly, since the number of relevant comparisons is restricted to the pairs formed by the selected person and the remaining members of the group. Despite the interpretative differences, both problems share a similar mathematical foundation based on combinatorial analysis and probability theory. In addition to the theoretical development, numerical results and graphical representations are presented to illustrate the behavior of the probabilities involved and to visualize how the likelihood of coincidences increases as the number of people grows. Finally, applications related to computing and cryptography are discussed, particularly the study of collisions in hash functions and the so-called birthday attack. The discussion highlights the importance of probabilistic reasoning in understanding apparently unlikely phenomena, while also emphasizing how the growth in the number of possible combinations directly influences the results obtained.
ER -