In a group of only 23 people, there is a 50% chance that two of them share the same birthday.

The Birthday Paradox is a classic problem in probability theory that highlights how human intuition often fails when dealing with exponential growth. While there are 365 days in a year, you do not need 183 people to reach a 50% probability of a match. Instead, the math relies on the number of possible pairs within the group. Using the combination formula nCr, a group of 23 people results in (23 × 22) / 2, which equals 253 unique pairs.To calculate the probability, mathematicians often look at the 'complementary event,' which is the probability that no one shares a birthday. For the first person, the probability is 365/365. For the second, it is 364/365, and so on. When you multiply these probabilities for all 23 people, the chance that everyone has a unique birthday drops to about 49.3%. This leaves a 50.7% chance that at least one pair shares a birthday.This phenomenon was famously discussed by Richard von Mises in 1939. It has significant implications in modern cryptography, specifically in 'birthday attacks' used to find collisions in hash functions. As the group size increases to 70 people, the probability of a shared birthday rises to a staggering 99.9%. The paradox demonstrates that while a match for a specific person is rare, a match between any two people in a group is highly likely due to the rapid accumulation of possible pairings.