16. Line-up. At the physical education teacher's whistle, all 10 boys and 7 girls lined up in a random order - wherever they managed to get. Find the mathematical expectation of the quantity "The number of girls standing to the left of all boys."
Solution. Let's mentally add one more boy, placing him to the right of everyone. Then the entire row is divided into 11 groups, each ending with a boy. The smallest group consists of one boy, and the largest can consist of 8 people - 7 girls and one boy.

Let's number these groups from left to right and call $\xi_{k}$ the size of the $k$-th group. We are interested in the expected size of the first group, that is, $\mathrm{E} \xi_{1}$.

The sum of all $\xi_{k}$ values is 18 (17 real students and one imaginary boy):

$$
\xi_{1}+\xi_{2}+\ldots+\xi_{11}=18
$$

Due to randomness, all $\xi_{k}$ values are identically distributed, so their expectations are equal to each other and equal to $\mathrm{E} \xi_{1}$. Transitioning to expectations in equation (2), we get: $11 \mathrm{E} \xi_{1}=18$, from which $\mathrm{E} \xi_{1}=\frac{18}{11}$. The expected number of girls is one less: $\frac{7}{11}$.

Answer: $\frac{7}{11}$.