What is the probability that in a group of 30 people, there are two people who share the same birthday?
We give the probability of the complementary event. For each person's birthday, there are 365 possibilities (we ignore the leap year). The birthdays of 30 people can be realized in $365^{30}$ ways. Now let's look at how many possibilities there are where any two people were born on different days. The first person could have been born on any of the 365 days. The second person could only be born on 364 days, i.e., not on the day the first was born. For the third person, there are only 363 possibilities, and so on. Thus, the probability is:

$$
\frac{365 \cdot 364 \cdot 363 \ldots 337 \cdot 336}{365^{30}} \approx 0.294
$$

and the probability of the original event is 0.706.

Rapai Tibor (Budapest, József A. Gymn.,)