Question:
- You have twelve friends. What is the probability none of them has their birthday in January?
- You have 365 friends in your social network. What is the probability that none of them have their birthday on 1 January?
- 35%. Each friend has their birthday in one of the other 11 months, i.e. assuming uniformly distributed birthday amongst 12 months $(\frac{11}{12})^{12} \approx 0.352$ or uniformly distributed birthdays amongst 365 days $(\frac{365-31}{365})^{12} \approx 0.348$.
- 36.7%. Each friend has their birthday on one of the other 364 days, $(\frac{364}{365})^{12} \approx 0.367$.
The answers are almost similar because both are part of the sequence $(\frac{n-1}{n})^n$ which converges to $1/e \approx 0.368$; see List of representations of e.