Example 3 Color each vertex of a regular nonagon with one of the two colors, red or blue, with equal probability. Then the probability that there exist three vertices of the same color forming an acute triangle is $\qquad$ .
【Analysis】If the convex polygon formed by vertices of the same color contains the center of the circumcircle of the regular nonagon, then there exist three vertices of this color such that the triangle they form also contains the center. Thus, this triangle is an acute triangle. Conversely, if the vertices of a certain color contain the vertices of an acute triangle, then the convex polygon generated by such vertices contains the center of the circumcircle of the regular nonagon.

If the convex polygons generated by the red and blue vertices do not contain the center, then the vertices of these two colors each lie in a semicircle of the circumcircle. The only possible scenarios for this to happen are that the red points are four consecutive vertices or five consecutive vertices, with each possibility having 9 cases.
Therefore, the probability $P=1-\frac{18}{2^{9}}=\frac{247}{256}$.