What is the birthday paradox probability for 23 people?

**At what group size does $P(\text{match})$ reach 99%?** Using the complement: $$P(\text{no match}) = \prod_{k=0}^{n-1} \frac{365 - k}{365}$$ We want $P(\text{no match}) \leq 0.01$. Solving: $n \approx 57$ or $58$. For $n = 57$: $P(\text{match}) \approx 0.9901$ For $n = 60$: $P(\text{match}) \approx 0.9941$ **Real-world applications:** 1. **Birthday attack on hash functions:** A hash function with $m$ possible outputs has $50\%$ chance of collision after about $\sqrt{2m \ln 2} \approx 1.17\sqrt{m}$ random inputs. For SHA-256 (256-bit), thats about $2^{128}$ attempts — still computationally infeasible. 2. **Cryptographic nonces:** When generating random session IDs, the birthday paradox determines how many IDs you can generate before collision becomes likely. 3. **Database indexing:** If you insert random keys into a database, the birthday paradox determines when the first collision occurs. The "paradox" isnt a logical contradiction — its just that human intuition about exponential growth is poor. We think linearly ("23 out of 365 = 6%") when we should think about pairs: $\binom{23}{2} = 253$ pairs, each with a $1/365$ chance of matching.