>[!abstract] >The birthday paradox is the counterintuitive probability result showing that in a group of just 23 people, there is about a 50% chance that at least two share the same birthday. While it seems unlikely given 365 possible birthdays, the paradox arises because the number of possible pairwise comparisons grows rapidly ($\frac{n(n-1)}{2}$) for $n$ people. This means even small groups yield many opportunities for matches. The birthday paradox illustrates how human intuition struggles with combinatorial growth and probabilistic reasoning, and it underlies applications in cryptography, particularly in hash collision analysis. >[!related] >- **North** (upstream): [[Probability theory]], [[Veridical paradox]] >- **West** (similar): [[Monty Hall problem]], [[Three prisoners problem]] >- **East** (different): [[Intuitive reasoning]] >- **South** (downstream): [[Collision probability]]