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Suppose a certain drug test is 99% accurate, that is, the test will correctly identify a drug
user as testing positive 99% of the time, and will correctly identify a non-user as testing negative 99%
of the time. This would seem to be a relatively accurate test, but Bayes's theorem will reveal a
potential flaw. Let's assume a corporation decides to test its employees for opium use, and 0.5% of the
employees use the drug. You want to know the probability that, given a positive drug test, an
employee is actually a drug user.
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Let assume that we have a 100,000 population of employees, and 0.5%, or 500 of them, are the drug users,
while the rest 100,000-500 = 99,500 are not the drug users.
Regarding these 99,500 employees, the test makes the error in 1% cases, i.e. the test mistakenly identifies
995 as the drug users, while they are not.
Regarding the disjoint set of 0.5% = 500 drug users, the text correctly identifies 99% of them, i.e. 495, as the drug users.
So, for now, we have, in all, 995+495 = 1490 persons identified as the grug users.
These 1490 employees are "the given set of positive drug tests".
Of them, 495 are true drug users.
So, the conditional probability under the problem's question is
= = 0.3322 (rounded), or 33.22%. ANSWER
Solved.
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It is, probably, a general rule:
when a test, which is designed for a large population and seems (is proclaimed) to be quite accurate,
is targeted/(is applied) to distinct a very narrow part of the entire population, then the test ceases to be so accurate.
Simply, the true measurement unit of the test precision should be
not in the parts of whole population, but in parts of this narrow sub-set.