6. Public rental housing sources in a city are located in neighborhoods $A$, $B$, and $C$. Each applicant can only apply for a house in one of these neighborhoods, and applying to any of the neighborhoods is equally likely. Then, for any 4 people applying in the city, the probability that exactly 2 people apply for housing in neighborhood $A$ is
Answer $\frac{8}{27}$.
Analysis If 4 people apply for houses in 3 neighborhoods, there are $\left(C_{3}^{1}\right)^{4}$ ways in total. If exactly 2 people apply for the house in neighborhood $A$, then the other 2 people can only apply for houses in neighborhoods $B$ or $C$, which gives $C_{4}^{2} C_{2}^{1} C_{2}^{1}$ ways. Therefore, the probability that exactly 2 people apply for the house in neighborhood $A$ is
$$
\frac{C_{4}^{2} C_{2}^{1} C_{2}^{1}}{\left(C_{3}^{1}\right)^{4}}=\frac{\frac{4 \times 3}{2} \times 2 \times 2}{3^{4}}=\frac{8}{27}
$$

Thus, the answer is $\frac{8}{27}$.