Question

An example of the use of Bayes' theorem is evaluation of drug test results. Suppose a certain drug test is 98% sensitive and 91% specific, that is, the test will correctly identify a drug user as testing positive 98% of the time and will correctly identify a non-user as testing negative 91% of the time. This would seem to be a relatively accurate test, but Bayes' theorem can be used to demonstrate the relatively high probability of misclassifying non-users as users. Let's assume a corporation decides to test its employees for drug use, and that only 7% of the employees actually use the drug. What is the probability that, given a positive test, an employee is actually a drug user?

Answer 1

Using Bayes' theorem, we calculate the probability of an employee being a drug user given a positive test result, with a sensitivity of 98% and specificity of 91%, and drug use prevalence of 7%. The result is approximately 44.19%, indicating a lower than expected certainty for positive test results.

Step-by-step explanation:

Using Bayes' theorem, we can calculate the probability that an employee who tested positive is actually a drug user. To do this, we'll denote U as the event that an employee uses drugs and T+ as the event that an employee tests positive for drugs. We're given that the sensitivity of the test (P(T+|U)) is 98%, the specificity (P(T-|U')) is 91%, and the prevalence (P(U)) of drug use is 7%. Bayes' theorem states:

P(U|T+) = [P(T+|U) * P(U)] / [P(T+|U) * P(U) + P(T+|U') * P(U')]

First, we find the probability of a non-user testing positive, P(T+|U'), which is the complement of the specificity, 100% - 91% = 9%. The probability that an employee is a non-user is the complement of the drug use prevalence, P(U') = 100% - 7% = 93%.

Now we can calculate:

P(U|T+) = (0.98 * 0.07) / (0.98 * 0.07 + 0.09 * 0.93) = 0.0686 / (0.0686 + 0.0867) ≈ 0.0686 / 0.1553 ≈ 0.4419 or 44.19%

Even with a test that appears accurate, the probability that an employee who tested positive is actually a drug user is roughly 44.19%.