## 11. How Many Are We?

To help you answer this question, let's say that the probability that at least two of us have the same birthday is less than $1 / 2$, but this would be false if one more person joined us.
11. The probability that two people do not share the same birthday is $364 / 365$. For three people, the probability that no two of them share the same birthday is (364/365)$\cdot$(363/365), and so on. In the case of $n$ people, the same probability is

$$
(364 / 365) \cdot(363 / 365) \cdot \ldots \cdot(365-n+1) / 365
$$

This product crosses the threshold of $1 / 2$ when $n$ changes from 22 to 23. Therefore, we are - 22 people.