生日悖论

先算补事件：23 人生日都不相同。

$$P(\text{全不同})=\frac{365}{365}\cdot\frac{364}{365}\cdot\frac{363}{365}\cdots\frac{343}{365}.$$

因此

$$P(\text{至少两人同生日})=1-P(\text{全不同})\approx 0.5073.$$

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Original Explanation

Instead of calculating the probability directly that at least two people share a birthday, we first calculate the complement: the probability that no one shares a birthday — that is, all birthdays are distinct — and subtract it from 1.

Step-by-step logic:

1. The first person can have any birthday:
   $$\frac{365}{365} = 1$$

2. The second person must have a different birthday than the first:
   $$\frac{364}{365}$$

3. The third person must differ from the first two:
   $$\frac{363}{365}$$

4. And so on, until the 23rd person:
   $$\frac{343}{365}$$

So, the probability that all 23 people have distinct birthdays is:

$$P(\text{distinct}) = \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \cdots \times \frac{343}{365}$$

Then, the probability that at least two people share a birthday is:

$$P(\text{shared}) = 1 - P(\text{distinct}) \approx 1 - 0.4927 = 0.5073$$

✅ Final Answer:

$$\boxed{0.5073}$$

So, in a room of 23 people, there is about a 50.73% chance that at least two people share the same birthday.