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I don't have a mathematical background so I apologise if this seems a bit simple for this forum, I'm not sure if this problem has a name in maths circles or is just too simple but I don't know the technical terms to search under.

I have a population of 3.2 billion and I want to measure something in X percent. I can sample up to half of the population at any one time. How would I calculate how many samples I would need to take to have measured X percent of my population. I presume there will be a mean number of samples I would have to measure and a standard deviation associated with this mean. I imagine if I was to plot n-samples, vs % of population sampled that it would converge towards 1.

I'm sure this is a simple problem for quant-people? Many thanks

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    $\begingroup$ Could you elaborate on what you mean by "measure something in X percent"? Why doesn't that just imply you should sample $X\%$ of the population? And what is the method of sampling? After all, in most circumstances where half the population can be sampled at one time, in two times the entire population can be sampled, so sampling only half would seem to be no limitation at all. $\endgroup$ – whuber Oct 18 '17 at 14:08
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    $\begingroup$ Do you mean that after you sample them you put them back and have no way of knowing who was sampled the first time, and so on? $\endgroup$ – mdewey Oct 18 '17 at 14:15
  • $\begingroup$ Your question doesn't seem to state what sort of sampling you're doing -- if resampling some population but my samples are not completely at random, I might mostly be sampling the same half each time. $\endgroup$ – Glen_b Oct 18 '17 at 23:16
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I interpret this question as asking about the situation that @mdewey described in the comments, that is, half of the units are sampled each time and then replaced, so that units can and will be sampled multiple times.

In that case, you can calculate this probability using the cumulative distribution function (CDF) of a Geometric distribution with parameter $p=0.5$. enter image description here

This shows that, for example, in order to cover at least 95% of the population on average, you'd need to take 5 samples.

If you want more refined probabilistic statements about the chances of including X% of the population in one of the samples (e.g., you want to know whether you have a 90% chance of covering 95% of the population, not just whether the expected proportion of sampled units exceeds 95%), then you can use the Geometric CDF value from above as the parameter $p$ of a Binomial distribution.

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  • $\begingroup$ This is very helpful - I imagined this exact plot in my head when I was thinking about the problem but didn't know it was called a CDF, so this is the technical term I was missing. $\endgroup$ – user3062260 Oct 19 '17 at 10:18
  • $\begingroup$ I think interpreting the problem as sampling and then putting them back is close enough simple analogy. Another way of thinking of it would be taking many copies of the entire population (genome sequences from many cloned cells) and being able to read the information from only half of the population in any one instance (being able to sequence only about half of the genome from each cell), how many cells would we need to sequence the whole genome. $\endgroup$ – user3062260 Oct 19 '17 at 10:18
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After the first sample, half the population will be unsampled. Looking at that set of unsampled members, on average half of them will not be sampled in the second iteration, leaving one fourth of the total population unsampled (one half of the one half). After n samples, (1/2)n of the population will remain unsampled. There will be some random variation in this number, but it will be on the order of the square root of the population size, which is this case gives ~50,000. So if you're okay with 1/k of the population not being sampled, then once 2n is significantly larger than k, you'll almost certainly have sampled your minimum. For instance, suppose you want 99% to be sampled. In other words, you're okay with 1 out of 100 being unsampled. Then since 27 = 128 > 100, taking 7 samples will be enough almost all of the time.

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  • $\begingroup$ I think this is more or less what I wanted, just to clarify the problem for other contributers; the 'population' is all the bases in a human genome sequence (3.2 billion), I am looking to sequence single cells and each time I run a sequence I cover about half of the genome, I needed to know how many cells to sequence to cover most of the genome. $\endgroup$ – user3062260 Oct 18 '17 at 15:30
  • $\begingroup$ @Jake Westfall I don't see how my answer depends on replacement. If there were no replacement, you would have all of the population sampled after two samples. Imagine 16 blue cubes. You randomly choose 8 to paint red. You now have 8 blue cubes and 8 red cubes. You now randomly choose 8 cubes to paint red again. On average, 4 of them will already be red, and 4 will be blue (there will be some variation, but on the scale of several billion, it will be very small). So you will now have 4 blue cubes and 12 red cubes. Etc. $\endgroup$ – Acccumulation Oct 18 '17 at 16:00

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