6. As shown in Figure 2, the coordinates of the four vertices of rectangle $O A B C$ are $(0,0), (2 \pi, 0), (2 \pi, 2), (0,2)$, 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
6. $\frac{1}{2}$.

Solution 1 Note that, the area of the rectangle $S=4 \pi$, the area of region $\Omega$ is
$$
\begin{array}{l}
S^{\prime}=4 \pi-\int_{0}^{2 \pi}(1+\cos x) \mathrm{d} x \\
=4 \pi-\left.(x+\sin x)\right|_{0} ^{2 \pi}=2 \pi .
\end{array}
$$

Therefore, the required probability $P=\frac{S^{\prime}}{S}=\frac{1}{2}$.
Solution 2 Using the method of cutting and complementing.
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 area of the rectangle. Hence, the required probability $P=\frac{1}{2}$.