The Birthday Collision

It is easier to calculate the probability that every birthday is different.

Multiply $365/365\times 364/365\times \cdots$ until the product falls below $1/2$.

1. 1. Use the complement

   Compute the chance that all birthdays are distinct.

2. 2. Form the product

   For $n$ people, the no-collision probability is a descending product.

   $$P(\text{no match})=\prod _{i=0}^{n-1}\frac{365-i}{365}$$

3. 3. Find the crossing

   At $n=22$ the product is about $0.524$; at $n=23$ it is about $0.493$. Therefore $23$ is the first group size above $50\%$.

Answer: 23