Any expression for the probability of a hard sphere in Boolean model I am working hard on a problem of Boolean model. In a example of Boolean model, points are scattered in the plane according to a homogeneous Poisson process of intensity λ. On each of these points a disc of fixed radius r is placed. 
My question is: What is the probability if a new disc of radius rh is placed without any contact of the existing discs in this Boolean model? 
 A: I'll add this as a second answer, as I'll take a different approach.
Here we do not make any requirement about the overlap of existing disks. While this satisfies the definition of a homogeneous Poisson (point) process, it conflicts with the requirements of a "hard sphere model".

We require that there is no contact between the new disk (red) and any existing disks (grey). This is equivalent to saying there there are zero points within the red/pink circle.
If this area is \begin{equation}A=π(r+rh)^2\end{equation}
and the Poisson intensity is λ. The number of points within this area is given by
\begin{equation}N \sim Po(\lambda A )\end{equation}
So
\begin{equation}P(No\ \ overlap) = P(N = 0)\end{equation}
\begin{equation}=\frac{(\lambda A)^0 e^{-\lambda A}}{0!}\end{equation}
\begin{equation}=e^{-\lambda A}\end{equation}
\begin{equation}=e^{-\lambda π(r+rh)^2}\end{equation}
For low densities, we can use the approximation
\begin{equation}e^{-\lambda π(r+rh)^2} = 1- \lambda π(r+rh)^2\end{equation}
Which is equal to the area available to place the centre of the new disk, after subtracting the "exclusion areas" for λ existing disks. (see my other answer)
A: I can't comment yet, so apologies for an incomplete answer.
Can you assume the existing discs do not overlap, being hard spheres?
If so, the probability the new disc does not overlap is related to the area left over after subtracting the "exclusion area" of each existing disc.
\begin{equation}
\pi( r + rh)^2
\end{equation}
To illustrate, new disc, shown in red.

If you reduce the new disc to a point (it's centre). The probability is related to area left after subtracting the exclusion areas (shown in grey), where the centre cannot go. We are assuming, the existing discs do not overlap, as per hard sphere model.

Finally, we need to consider what happens if the existing spheres are close enough for the "exclusion areas" to overlap. This can be ignored as negligible if the existing spheres are widely spaced. If, however, they are close, this becomes very significant.

