Statistics Concept — The Birthday Paradox: A Conditional Probability Perspective
Last Updated on June 3, 2026 by Editorial Team
Author(s): Chao De-Yu
Originally published on Towards AI.
Why shared birthdays emerge much earlier than intuition suggests — via conditional probability, interactions, and Monte Carlo simulation in Python
In probability, some results feel so counterintuitive that they almost seem wrong at first glance. The Birthday Paradox is one of the most famous examples.
After the introduction, the article lays out the Birthday Paradox formally (365 days, uniform and independent birthdays) and solves it via the complement: instead of computing the chance of “at least one shared birthday” directly, it computes the probability that a new person avoids all previous birthdays. Using the chain rule, it turns the problem into a product of conditional probabilities (365/365, then 364/365, 363/365, etc.), derives the result for 23 people (~0.5073), and explains why intuition fails by reframing the situation as a sequence of dependent conditions and, alternatively, as the rapid growth of pairwise comparisons (~nC2). Finally, it confirms the theory with a Monte Carlo simulation in Python, then concludes that the paradox is driven by combinatorial interaction growth, not by any single person’s “small” mismatch chance.
Read the full blog for free on Medium.
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