A test for a certain drug produces a false negative 5% of the time and a false positive 8% of the time. Suppose 12% of the employees at a certain company use the drug. If an employee at the company tests positive, what is the probability that he or she does not use the drug?
To find the probability that an employee does not use the drug given that they tested positive, we can use Bayes' theorem.
Let's define the events:
D = Employee uses the drug
D' = Employee does not use the drug
P = Employee tests positive
We are given that P(D) = 0.12 (probability that an employee uses the drug) and P(P|D') = 0.08 (probability of a false positive).
We want to find P(D'|P), the probability that an employee does not use the drug given that they tested positive.
According to Bayes' theorem:
P(D'|P) = (P(D') * P(P|D')) / P(P)
To find P(P), the probability of testing positive, we need to consider both scenarios:
1) Employee uses the drug and tests positive (true positive).
2) Employee does not use the drug and tests positive (false positive).
The probability of testing positive can be calculated as:
P(P) = P(D) * P(P|D) + P(D') * P(P|D')
We are given P(P|D') = 0.08 (probability of a false positive) and P(D') = 1 - P(D) = 1 - 0.12 = 0.88.
Substituting the given values, we have:
P(P) = 0.12 * P(P|D) + 0.88 * 0.08
Now we can substitute the values into Bayes' theorem:
P(D'|P) = (0.88 * 0.08) / (0.12 * P(P|D) + 0.88 * 0.08)
Suppose we assume that the test is 100% accurate (no false negatives), then P(P|D) would be 1 (since if an employee uses the drug, they would definitely test positive).
P(D'|P) = (0.88 * 0.08) / (0.12 * 1 + 0.88 * 0.08)
Calculating this expression will give us the probability that an employee does not use the drug given that they tested positive.