Assuming I observe, in a unit square, $n_1$ circles of area $A_1$ (non-overlapping amongst themselves) and $n_2$ circles of area $A_2$ (again, non-overlapping) and that each of the centres is uniformly distributed across the square (well, I guess that can't be precisely true given the non-overlapping constraint), is there an expression for the proportion of the first circles that overlap with at least one of the second type of circles.
The actual situation is observed cancer cells and pores on a microscope slide, to which the above seems a good approximation.
In the 1D case, I'd like to think that for each line segment (1D equivalent of a circle) of the first group, there's an interval of size $A_1+ A_2$ before the end of each segment of the second type to place the start of the interval so that they would overlap, and so $n_2(A_1+A_2)$ of the unit interval would result in an overlap. That seems to be true for the first segment of the first group, but it won't necessarily hold for further segments as the non-overlapping criterion breaks the independence, so I'm probably looking at this (and the 2D case) from the wrong angle.
Updated thoughts
Maybe this is the way to look at it? There's a total area $n_1A_1$ covered by type 1, and $n_2A_2$ covered by type 2, out of a total area of 1. So an expected area of $n_1A_1n_2A_2$ is covered by both, and divide this by $A_1$ to give the number of type 1's that would be needed to cover that area, and divide by $n_1$ to get the proportion of type 1's that are in the double-covered area. So if 7% of a slide's area is covered by spores, then 7% of cells are hit by spores?