13. There are fewer than 30 people in the class. The probability that a randomly chosen girl is an excellent student is $\frac{3}{13}$, and the probability that a randomly chosen boy is an excellent student is $\frac{4}{11} \cdot$ How many excellent students are there in the class?
# Solution.

According to classical probability theory, the probability that a randomly chosen girl is an excellent student is equal to the ratio of the number of excellent girl students to the total number of girls in the class. Accordingly,

$$
\frac{3}{13}=\frac{\text { number of girls-excellent students }}{\text { number of girls }}
$$

Considering that the number of girls is a natural number, and that there are fewer than 30 people (and thus fewer than 30 girls) in the class, we find that there are either 13 girls (3 excellent students) or 26 (6 excellent students).

Applying the same reasoning to boys, we find that:

$$
\frac{4}{11}=\frac{\text { number of boys-excellent students }}{\text { number of boys }}
$$

In the class, there are either 11 boys (4 excellent students) or 22 (8 excellent students). Further, considering that there are fewer than 30 people in the class, we find that there are 13 girls (3 excellent students) and 11 boys (4 excellent students). Therefore, the number of excellent students is $3+4=7$.

Answer: 7.