生日悖论：超过 1/2 需要多少人

经典结果是 23 人。

令 $n$ 为人数。补事件为“生日全不同”，其概率为

$$\frac{365\cdot 364\cdots (365-n+1)}{365^n}.$$

当 $n=23$ 时，$1-\text{上述概率}>1/2$，且这是最小满足条件的 $n$。

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Original Explanation

The classic result is 23. Specifically, for $n$ people, the total number of ways to assign birthdays is $365^n$. The number of ways to assign distinct birthdays to all $n$ is $$365 \times 364 \times \cdots \times (365 - n + 1).$$ We need $$1 \;-\; \frac{ 365 \times 364 \times \cdots \times (365 - n + 1) }{ 365^n } \;>\; \frac{1}{2},$$ which first holds at $n=23$.