Problem Analysis #3

To find the probability $p_9$ that a randomly chosen quadrilateral formed by four distinct vertices of a regular $9$-gon contains the center of the polygon in its interior, we can use the complement.

The total number of ways to choose $4$ vertices from $9$ is:

A quadrilateral does not contain the center of the polygon in its interior if and only if all of its $4$ vertices lie within some semi-circle. Since $n=9$ is odd, the maximum number of vertices that can lie in a semi-circle is $\frac{9-1}{2} = 4$ (excluding the starting vertex).

Using the property of circular intervals, the number of subsets of size $k$ whose vertices all lie within a semi-circle for an odd $n$-gon is given by:

For $n=9$ and $k=4$, the number of quadrilaterals that do not contain the center is:

Thus, the number of quadrilaterals that do contain the center in their interior is:

The probability $p_9$ is therefore:

This fraction is in lowest terms, so $p = 5$ and $q = 7$. We find:

\boxed{12}