20. The number of staff and the distribution of male and female staff in two departments, A and B, of a certain unit are as follows. Now, a stratified sampling method (using simple random sampling without replacement within strata) is adopted to select 3 staff members from departments A and B to conduct a survey on "low-carbon living".
(1) Find the number of people to be sampled from departments A and B;
(2) Find the probability that at least 1 female staff member is selected from department A;
(3) Let $\xi$ represent the number of male staff members among the 3 selected. Find the distribution of $\xi$ and its expected value.
(1) The sampling ratio is 5:1, so 2 people are drawn from Department A, and 1 person is drawn from Department B.
(2) Let $A=$ “at least 1 female is drawn from the staff of Department A”, then $P(A)=1-\frac{C_{6}^{2}}{C_{10}^{2}}=1-\frac{15}{45}=\frac{2}{3}$.
(3) $\xi=0,1,2,3$, and $P(\xi=0)=\frac{C_{4}^{2} \cdot C_{2}^{1}}{C_{10}^{2} \cdot C_{5}^{1}}=\frac{4}{75}, P(\xi=1)=\frac{C_{6}^{1} \cdot C_{4}^{1} \cdot C_{2}^{1}+C_{4}^{2} \cdot C_{3}^{1}}{C_{10}^{2} \cdot C_{5}^{1}}$
$$
=\frac{22}{75}, P(\xi=2)=\frac{C_{6}^{2} \cdot C_{2}^{1}+C_{6}^{1} \cdot C_{4}^{1} \cdot C_{3}^{1}}{C_{10}^{2} \cdot C_{5}^{1}}=\frac{34}{75}, P(\xi=3)=\frac{C_{6}^{2} \cdot C_{3}^{1}}{C_{10}^{2} \cdot C_{5}^{1}}=\frac{1}{5} .
$$
The distribution of $\xi$ is as follows:
The expectation of $\xi$ is $E \xi=\frac{22+2 \cdot 34+3 \cdot 15}{75}=\frac{9}{5}$.