Simulation of the Second Matern Hard Core point process

Question

Using the second matern point process thinning method, I have obtained a hard core point process from a regular poisson process in $\mathbb{R}^2$. According to the standard text (Stochastic Geometry and its Applications), the new intensity of the thinned point process is \begin{equation} \lambda = \frac{1 - exp(-\lambda_{PP}\pi r^2)}{\pi r^2} \end{equation} where, $\lambda_{PP}$ represents the intensity of the original Poisson process and $r$ is the hard core distance between the points. I understand that $\lambda$ signifies the $\textbf{expected number of points in a region of unit area}$. I want to compute the probability of having at-least one point in a region of area $A_r$. Treating this like a poisson process, I get \begin{equation} \phi = 1 - exp(-\lambda A_r) \end{equation} I have been unable to verify this probability calculation using simple simulations. The probability estimate obtained after many iterations is significantly higher than the one predicted by the above formula. Any idea why this might be happening?