(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 $\qquad$.
6. $\frac{1}{2}$ Hint: Method 1 The area of the rectangle $S=4 \pi$, the area of region $\Omega$
$$
S^{\prime}=4 \pi-\int_{0}^{2 \pi}(1+\cos x) \mathrm{d} x=4 \pi-\left.(x+\sin x)\right|_{0} ^{2 \pi}=2 \pi \text {. }
$$

Thus, the required probability $P=\frac{S^{\prime}}{S}=\frac{1}{2}$.
Method 2 Using the cut-and-paste method, it is easy to see that the area of region $\Omega$ is equal to the area of the "blank" part inside the rectangle, i.e., the area of region $\Omega$ is $\frac{1}{2}$ of the rectangle's area.
Thus, the required probability $P=\frac{1}{2}$.