The Birthday Paradox

In a group of 23 randomly chosen people, the probability that two share a birthday exceeds 50% – surprising because 23/365 ≈ 6.3%. The paradox arises because we compare every pair, not a fixed target. With n people, number of pairs is n(n‑1)/2. For n=23, that’s 253 pairs. The probability no pair shares a birthday is product (364/365)·(363/365)… ≈ 0.4927, so at least one shared birthday ≈ 0.5073. For n=57, probability > 99%. This principle is used in cryptography (birthday attacks on hash functions).