20. (12 points) The number of staff and the distribution of male and female staff in two departments, A and B, of a certain unit are shown in Table 1. Now, a stratified sampling method (using simple random sampling without replacement within strata) is adopted to select three staff members from the two departments for a survey on "low-carbon living".
(1) Determine the number of people to be sampled from departments A and B;
(2) Find the probability that at least one female staff member is selected from department A;
(3) Let $\xi$ represent the number of male staff members among the three selected. Find the distribution of $\xi$ and its expected value.
20. (1) The number of people to be drawn from departments A and B are $3 \times \frac{10}{15}=2$ and $3 \times \frac{5}{15}=1$, respectively.
(2) Let event $A$ be: at least one female is drawn from department A.
Then $P(A)=1-\frac{\mathrm{C}_{6}^{2}}{\mathrm{C}_{10}^{2}}=\frac{2}{3}$.
(3) The possible values of $\xi$ are $0, 1, 2, 3$, then
$$
\begin{array}{c}
P(\xi=0)=\frac{\mathrm{C}_{4}^{2}}{\mathrm{C}_{10}^{2}} \cdot \frac{\mathrm{C}_{2}^{1}}{\mathrm{C}_{5}^{1}}=\frac{4}{75}, \\
P(\xi=1)=\frac{\mathrm{C}_{4}^{1} \mathrm{C}_{6}^{1}}{\mathrm{C}_{10}^{2}} \cdot \frac{\mathrm{C}_{2}^{1}}{\mathrm{C}_{5}^{1}}+\frac{\mathrm{C}_{4}^{2}}{\mathrm{C}_{10}^{2}} \cdot \frac{\mathrm{C}_{3}^{1}}{\mathrm{C}_{5}^{1}}=\frac{22}{75}, \\
P(\xi=3)=\frac{\mathrm{C}_{6}^{2}}{\mathrm{C}_{10}^{2}} \cdot \frac{\mathrm{C}_{3}^{1}}{\mathrm{C}_{5}^{1}}=\frac{1}{5}, \\
P(\xi=2)=1-P(\xi=0)-P(\xi=1)-P(\xi=3)=\frac{34}{75} . \\
\text { Hence } \mathrm{E} \xi=\sum_{k=0}^{3} \xi P(\xi=k) \\
=0 \times \frac{4}{75}+1 \times \frac{22}{75}+2 \times \frac{34}{75}+3 \times \frac{1}{5}=\frac{9}{5} .
\end{array}
$$