23. (2004 National College Entrance Examination, Fujian Province) Two people, A and B, participate in an English test. It is known that among the 10 questions available, A can answer 6 of them, and B can answer 8 of them. It is stipulated that each test will randomly select 3 questions from the available questions for testing, and at least 2 questions must be answered correctly to pass.
(I) Find the probability distribution and mathematical expectation of the number of questions A answers correctly, $\xi$;
(II) Find the probability that at least one of A and B passes the test.

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23. (I) The probability distribution of the number of questions $\xi$ that person 甲 answers correctly is as follows:
\begin{tabular}{|c|c|c|c|c|}
\hline$\xi$ & 0 & 1 & 2 & 3 \\
\hline$P$ & $\frac{1}{30}$ & $\frac{3}{10}$ & $\frac{1}{2}$ & $\frac{1}{6}$ \\
\hline
\end{tabular}
$$
E \xi=0 \times \frac{1}{30}+1 \times \frac{3}{10}+2 \times \frac{1}{2}+3 \times \frac{1}{6}=\frac{9}{5} .
$$
(II) Let the events that 甲 and 乙 pass the exam be $A$ and $B$, respectively, then $P(A)=\frac{C_{6}^{2} C_{4}^{1}+C_{5}^{3}}{C_{10}^{3}}=\frac{2}{3}, P(B)=\frac{C_{8}^{2} C_{2}^{1}+C_{8}^{3}}{C_{10}^{3}}=$ $\frac{14}{15}$.
Since events $A$ and $B$ are independent,
Method 1: The probability that both 甲 and 乙 fail the exam is
$$
P(\bar{A} \cdot \bar{B})=P(\bar{A}) \cdot P(\bar{B})=\left(1-\frac{2}{3}\right)\left(1-\frac{14}{15}\right)=\frac{1}{45} \text {. }
$$

Therefore, $P=1-P(\bar{A} \cdot \bar{B})=\frac{44}{45}$.
Method 2: The probability that at least one of 甲 and 乙 passes the exam is
$$
P=P(A \cdot \bar{B})+P(\bar{A} \cdot B)+P(A \cdot B)=\cdots=\frac{44}{45} .
$$