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Given a surface with area $A$, what is the expected area of the region occupied by $k$ possibly overlapping random circles with equal radii $r$?

For example, I would like to estimate the area of the black region of this simulation.

If it helps, we can assume the edges of surface wraps around, so the surface is a torus. And instead of random circles, it can also be random squares.

I've tried to find an approximation by partitioning the space as a "pixel" grid and solve it as a K balls in N buckets problem. But I didn't find how to generalize it from balls to circles.

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    Hint: The expected area of the occupied area$C$is the integral over $x$ in $A$ of $P(x\in C)$. What is $P(x\in C)$?2012-04-25

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Making all the necessary assumptions to make this question nice, if a point has a probability $\frac{a}{A}$ of being covered by a small object of area $a$, then the probability of not being covered by it is $1-\frac{a}{A} = \frac{A-a}{A}$.

So the probability of not being covered by $k$ small independent objects is $\left(\frac{A-a}{A}\right)^k$ and the probability of being covered by at least one of them is $1- \frac{(A-a)^k}{A^k}$.

So the area covered is $A-\frac{(A-a)^k}{A^{k-1}}.$

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    @Nelson: $A$ is the total area, $a$ is the area of each of the small objects, and $k$ is the number of small objects2013-04-13