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A280989
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Rounded percent probability that a positive test result for some condition is a true positive when n% of the tested population has that condition and the test is 95% accurate.
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0
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0, 16, 28, 37, 44, 50, 55, 59, 62, 65, 68, 70, 72, 74, 76, 77, 78, 80, 81, 82, 83, 83, 84, 85, 86, 86, 87, 88, 88, 89, 89, 90, 90, 90, 91, 91, 91, 92, 92, 92, 93, 93, 93, 93, 94, 94, 94, 94, 95, 95, 95, 95, 95, 96, 96, 96, 96, 96, 96, 96, 97, 97, 97, 97, 97, 97, 97, 97, 98, 98, 98, 98, 98, 98, 98, 98, 98, 98, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 100, 100, 100, 100, 100, 100, 100, 100, 100
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OFFSET
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0,2
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COMMENTS
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"95% accurate" means the test result is correct 95% of the time symmetrically: the test failure rate is 5% when the test result should be positive and 5% when the result should be negative.
This perhaps-unexpected effect was demonstrated by Paulos with an example similar to the one below.
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REFERENCES
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John Allen Paulos, Innumeracy: Mathematical Illiteracy and Its Consequences, Hill and Wang, 1988.
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LINKS
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FORMULA
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a(n) = round(950*n/(9*n + 50)), 0 <= n <= 100.
In general, a(n, r) = round(100*r*n/(2*r*n - 100*r - 100*n + 10000)), 0 <= n <= 100, 0 <= r <= 100, where r is the accuracy rate in percent (r = 95 for this sequence).
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EXAMPLE
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If a test for a disease (that is relatively rare) is 95% accurate and is given once to a population of, say, 5000 individuals, when only 2% of them actually have the disease, the expected results are 245 (= 0.05 * 4900) false positives and 95 (= 0.95 * 100) true positives. Out of the 340 who test positive, 95 people have the disease and 95/340 = 19/68 = 0.279..., which rounds to 28%, so a(2) = 28 -- about 28% of the people who tested positive actually have the tested-for disease. On the other hand, only 5/4660 = 0.00107... or roughly 0.1% of the people who tested negative actually have the affliction.
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PROG
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(PARI) a(n) = if(n >= 0 && n <= 100, round(950*n/(9*n + 50)))
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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