Example 2 (2004 National College Entrance Examination, Hunan Province) Three machines, A, B, and C, independently process the same type of part. It is known that the probability of machine A processing a first-class part while machine B does not is $\frac{1}{4}$, the probability of machine B processing a first-class part while machine C does not is $\frac{1}{12}$, and the probability of both machines A and C processing first-class parts is $\frac{2}{9}$.
(1) Find the probability that each of the three machines A, B, and C processes a first-class part;
(2) One part is selected from those processed by machines A, B, and C for inspection. Find the probability that at least one of them is a first-class part.
Analysis We use the relationship between independent events and mutually exclusive events to set up equations and solve them.
Solution (1) Let $A, B, C$ be the events that the parts processed by machines Jia, Yi, and Bing are of first quality, respectively.
From the given conditions, we have
$$
\left\{\begin{array}{l}
P(A \cdot \bar{B})=\frac{1}{4}, \\
P(B \cdot \bar{C})=\frac{1}{12}, \\
P(A \cdot C)=\frac{2}{9},
\end{array}\right.
$$

which means
$$
\left\{\begin{array}{l}
P(A)(1-P(B))=\frac{1}{4}, \\
P(B)(1-P(C))=\frac{1}{12}, \\
P(A) P(C)=\frac{2}{9}, \\
\left\{\begin{array}{l}
P(A)=\frac{1}{3}, \\
P(B)=\frac{1}{4}, \\
P(C)=\frac{2}{3} .
\end{array}\right.
\end{array}\right.
$$

Therefore, the probabilities that the parts processed by machines Jia, Yi, and Bing are of first quality are $\frac{1}{3}, \frac{1}{4}, \frac{2}{3}$, respectively.
(2) Let $D$ be the event that at least one part is of first quality when one part is inspected from each of the parts processed by Jia, Yi, and Bing, then $P(D)=1-P(\bar{D})=1-(1-P(A))(1-P(B))(1-P(C))$
$$
=1-\frac{2}{3} \cdot \frac{3}{4} \cdot \frac{1}{3}=\frac{5}{6} .
$$

Therefore, the probability that at least one part is of first quality when one part is inspected from each of the parts processed by Jia, Yi, and Bing is $\frac{5}{6}$.
!Note This problem mainly examines the relationship between the concepts of independent events and mutually exclusive events.