Surprising Probability

In a room of just 23 people, there is a 50% chance that two share a birthday. With 50 people, it jumps to 97%. This is the birthday paradox. It feels impossible because we think about the chance of someone sharing our birthday, not the chance of any pair matching.

The math works because you compare every possible pair of people, not just one person against the rest. With 23 people there are 253 pairs, and each pair has a small chance of matching. Those small chances compound fast. The birthday paradox is not really a paradox at all. It is a failure of human intuition about how quickly pairwise comparisons add up.