(6) As shown in the figure, the coordinates of the four vertices of rectangle $O A B C$ are $(0,0)$, $(2 \pi, 0)$, $(2 \pi, 2)$, and $(0,2)$, respectively. Let the region (shaded in the figure) enclosed by side $B C$ and the graph of the function $y=1+\cos x(0 \leqslant x \leqslant 2 \pi)$ be $\Omega$. If a point $M$ is randomly thrown into rectangle $O A B C$, then the probability that point $M$ falls within region $\Omega$ is $\qquad$.