Since $\int_{0}^{\pi}(\cos x+1) \mathrm{d} x=\left.(\sin x+x)\right|_{0} ^{\pi}=\pi$, the area of the "heart" is $3 \pi$, and the area of $\Omega$ is $4(\pi+1)$, so the probability that $A$ is within the "heart" is $P=\frac{3 \pi}{4(\pi+1)}$.
8. As shown in the figure, the "heart" is a closed shape formed by the curves $C_{1}: x^{2}+y^{2}=2|y|(x \leqslant 0)$ and $C_{2}:|y|=$ $\cos x+1(0 \leqslant x \leqslant \pi)$,
If a point $A$ is randomly selected within the region $\Omega=\left\{(x, y) \left\lvert\,\left\{\begin{array}{l}-1 \leqslant x \leqslant \pi, \\ -2 \leqslant y \leqslant 2\end{array}\right\}\right.\right.$,
then the probability $P=$ $\qquad$ that $A$ is inside the "heart" is