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In an experiment I have a total surface to cover ($100\%$). Experiment runs are independent of each other and cover approximately $30\%$ of this surface. I am only interested in the total unique surface covered (as a $\%$).

I want to find out the formula that links the number of experiment runs to total unique surface covered.

I hope my title wasn't misleading but I didn't know how to concisely phrase my question.

Thank you.

Aditya Dev
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scart123
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    The title is indeed a bit misleading, as the body of the question seems to say that you're interested in the surface covered as a function of the number of runs, rather than the number of runs required to cover the entire surface. Another problem is that you don't say anything about how the $30%$ that are covered are selected; the answer will depend strongly on that. You may be interested in this question: http://math.stackexchange.com/questions/176383. – joriki Apr 11 '16 at 15:39

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Each experiment misses each particular point of the region with probability $0.7$, so $n$ experiments miss each particular point every time with probability $0.7^n$. Therefore $ 1- 0.7^n$ is the probable fraction covered after $n$ experiments. That approaches $1$ as $n$ grows. Choose your $n$ to give you an acceptable miss percentage.

This argument depends on two independence assumptions. One you state - the experiments are independent of each other. The second is implicit - the 30% coverage chooses points independently of each other. That's the issue @joriki raises in his comment. If your surface is a plane region of some sort and the experiment chooses a reasonably shaped subregion that may not be the case.

Ethan Bolker
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    This assumes that each run covers an almost surely non-measurable set in which each point is selected independently with probability $0.3$. I wonder whether this is what the OP had in mind. – joriki Apr 11 '16 at 15:42
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    @joriki See my edit in response to your comment. – Ethan Bolker Apr 11 '16 at 15:46