The Math Behind It

The easiest way to calculate the birthday probability is to compute the chance that nobody shares a birthday and subtract from 1. Person 1 can have any birthday (365/365). Person 2 must differ: 364/365. Person 3: 363/365. Multiply these fractions for all 23 people.

This means the complementary probability — at least one match — exceeds 50%. The calculation reveals that probabilities compound faster than we expect. Each new person doesn't just add one comparison; they add a comparison with every existing person in the room.