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PLANNING SCHOOLS ENVIRONMENT EROSION
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Random Samples from a Property Can Miss Problems
For example if you analyze 18 samples from 18 acres and find no arsenic, you still have a 10% chance of missing scattered arsenic that covers fraction .1201 of the site (2^{1}/_{6} acres out of 18). You'd have a 5050 chance of missing arsenic on 2/3 of an acre (.0378 of 18 acres). Your 18 samples have a 90% chance of missing pollution on fraction .0058 of the site (which is 1/9 of an acre, or over 4,000 square feet). With larger numbers of samples, the area you might miss gets smaller. For example with 500 samples, scattered arsenic totaling 3,600 square feet (.0046 of the 18 acres) would still have a 10% chance of being missed, and 165 square feet would have a 90% chance of being missed. This shows both the importance of large numbers of samples, and the need to know where sources of pollution are, since random samples are not very good at finding small concentrations of pollution. If the 500 samples are spread over 1,000 acres instead of 18 acres, then 200,000 square feet of arsenic scattered on the property would have a 10% chance of being missed, and 9,000 square feet would have a 90% chance of being missed. So when you hear, "We took samples and didn't find any ...", that doesn't mean there's no problem. All this text applies to DDT, lead, or any other pollutant that stays put in soils. We don't mean to pick on arsenic. 
If you take this many samples and find no problem in them 
Then you have a 





10% 
50% 
90% 






chance of missing a problem that's present in the following fraction of the land: 







1 
.9000 
.5000 
.1000 






10 
.2057 
.0670 
.0105 






18 
.1201 
.0378 
.0058 






20 
.1087 
.0341 
.0053 






30 
.0739 
.0228 
.0035 






40 
.0559 
.0172 
.0026 






50 
.0450 
.0138 
.0021 






60 
.0376 
.0115 
.0018 






70 
.0324 
.0099 
.0015 






80 
.0284 
.0086 
.0013 






90 
.0253 
.0077 
.0012 






100 
.0228 
.0069 
.0011 






200 
.0114 
.0035 
.0005 






300 
.0076 
.0023 
.0004 






400 
.0057 
.0017 
.0003 






500 
.0046 
.0014 
.0002 






600 
.0038 
.0012 
.0002 






700 
.0033 
.0010 
.0002 






800 
.0029 
.0009 
.0001 






900 
.0026 
.0008 
.0001 






1000 
.0023 
.0007 
.0001 

Methodology: Let P represent the proportion of the site that is contaminated (by some item, beyond some acceptable level; choose your poison). Then each sample has P probability of hitting the contamination, and 1P probability of missing it. n independent samples have (1P)^{n} chance that they all miss the contamination. Let C represent this chance that all samples miss the contamination, so C = (1P)^{n} Solving for P gives the formula P = 1C^{1/n} , which was used to calculate the numbers above. (This multiplication of independent probabilities assumes scattered contamination, such as loading stations. A systematic grid of 1 sample per acre would have a better chance of finding a continuous area of contamination, such as flooding by a contaminant, or air emissions from a neighboring smelter.) 