找到同生日的人

30: Finding Your Birthmate

你想找到一个与你生日相同的人。最少需要问多少个陌生人的生日，才能有 50% 的概率至少遇到一个与你同生日的人？（忽略闰年，假设一年 365 天等可能）

You want to find someone whose birthday matches yours.

What is the least number of strangers whose birthdays you need to ask about to have a 50–50 chance?

解析

任一陌生人生日不等于你的概率为 $\frac{364}{365}$ 。

问 $n$ 个人都不等于你的概率为 $\left(\frac{364}{365}\right)^n$ 。

要求至少 50% 概率“有人等于你”，即

\left(\frac{364}{365}\right)^n\le \frac12.

解得 $n\approx 253$ ，因此至少问 253 人。

The probability that a randomly chosen stranger does not share your birthday is:

\frac{364}{365}

The probability that $n$ strangers all do not share your birthday is:

\left( \frac{364}{365} \right)^n

To have at least a 50% chance of finding someone who does share your birthday, solve for the smallest $n$ such that:

\left( \frac{364}{365} \right)^n \leq 0.5

Taking logarithms:

n \cdot \log\left( \frac{364}{365} \right) \leq \log(0.5)

Solving gives:

n \geq \frac{\log(0.5)}{\log\left( \frac{364}{365} \right)} \approx 253

Answer:
You need to ask at least 253 people to have a 50-50 chance that one of them shares your birthday.