(a) What is the probability that a randomly selected Floridian welfare recipient uses illegal drugs and has a positive test?
(b) What is the probability that a randomly selected Floridian welfare recipient does not use illegal drugs but nevertheless has a positive test?
(c) What is the probability that a randomly selected Floridian welfare recipient has a positive test?
(d) Given that a randomly selected Floridian welfare recipient has a positive test, what is the probability that he or she uses illegal drugs?
(e) If a Florida voter favors the law because he thinks the answer to (d) is in the neighborhood of 90 percent, what fallacy might he be committing?
(a) Probability of drugs user with positive test: 0.072.
(b) Probability of non-drug user with positive test: 0.092.
(c) Probability of positive test: 0.164.
(d) Probability of drug user given positive test: 0.439.
(e) Fallacy: Neglecting base rate if assuming 90% drug user probability given positive test.
Calculating Probabilities in Mandatory Drug Testing for Welfare Recipients
(a) Probability of drugs user with positive test: To calculate this probability, we use Bayes' theorem. Considering an 8% drug use rate and a 90% accurate test, the probability is:
P(Drug user and Positive Test) = P(Drug user) * P(Positive Test | Drug user) = 0.08 * 0.9 = 0.072.
(b) Probability of non-drug user with positive test: This probability can be calculated as P(No Drug user and Positive Test) = P(No Drug user) * P(Positive Test | No Drug user). Assuming 92% are not drug users, the probability is:
P(No Drug user and Positive Test) = 0.92 * 0.1 = 0.092.
(c) Probability of positive test: The probability that a randomly selected Floridian welfare recipient has a positive test is the sum of the probabilities of having a positive test given drug use and having a positive test given no drug use.
P(Positive Test) = P(Drug user and Positive Test) + P(No Drug user and Positive Test) = 0.072 + 0.092 = 0.164.
(d) Probability of drug user given positive test: Using Bayes' theorem, we calculate the probability that a welfare recipient uses illegal drugs given a positive test.
P(Drug user | Positive Test) = (P(Drug user) * P(Positive Test | Drug user)) / P(Positive Test).
Using the calculated values, we get:
P(Drug user | Positive Test) = (0.08 * 0.9) / 0.164 = 0.439.
(e) Fallacy of Neglecting Base Rate: If a Florida voter believes the probability of a drug user given a positive test is around 90%, they might be committing the fallacy of neglecting the base rate. The actual probability calculated in (d) is only 44%, much lower than their assumption.