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The problem arose after a discussion why larger digital camera photo sensors is much more expensive than little bit smaller ones, and the reason was given that it's due to difficulty of finding a larger area spot on a big CCD or CMOS panel.

Consider a large clear white sheet (of a given area $S$, and we may consider it of any convenient non-degenerate shape, such as square or circle) with some black dirt dots on it. The average density of the dirt dots is uniform and known to be $p$ dots per unit area. Somebody wants to find a clear round spot of radius $r$ on it.

Question 1: how difficult is it to find such a spot (and the term "difficult" maybe is defined as "the probability of a random disc being clear?"). How many such non-overlapping spots there are on the sheet on average?

Question 2: how much more difficult is it to find a spot of radius $k\cdot r$ with $k \gt 1$ than a spot of a radius $r$?

When $S\gg s=\pi r^2$, this looks easy, but when $S$ is comparable to $s$, the result is not so obvious.

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I'll answer the first part of Question 1. I think the second part of Question 1 is probably intractable analytically; and Question 2 is just a special case of the first part of Question 1.

The number of spots on any given area is Poisson-distributed. On a spot of area $s$ with dot density $p$, the expected number of dots is $ps$, so the Poisson distribution is given by

$p(n) = \frac{(ps)^n\mathrm e^{-ps}}{n!}\;.$

The probability of the spot being clear is $p(0)=\mathrm e^{-ps}$. Thus, the reason given was correct; finding a clear spot becomes exponentially difficult with the spot's area $s$.

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    Yes, estimate based on area ratio is good for small number of dots, an very large sheets relative to spots. However, there are very "unfortunate" spot distributions that allows very few clear spots available (or even none). For example, a situation where dots are distributed in vertices of a triangle grid.2011-07-11