Jim is a randomly chosen member of a large population in which 6% are heroin users. Joe tests positive for heroin in a drug test that correctly identifies users 98% of the time and correctly identifies nonusers 95% of the time. What is the probability that Jim is a heroin user given that his test came back positive?
To find the probability that Jim is a heroin user given that his test came back positive, we can use Bayes' theorem. Bayes' theorem states that:
P(A | B) = (P(B | A) * P(A)) / P(B),
where P(A | B) is the probability of event A occurring given that event B has occurred, P(B | A) is the probability of event B occurring given that event A has occurred, P(A) is the probability of event A occurring, and P(B) is the probability of event B occurring.
In this case, event A is the event that Jim is a heroin user, and event B is the event that Jim's test came back positive. We are given the following information:
- P(A) = 0.06 (6% of the large population are heroin users).
- P(B | A) = 0.98 (the test correctly identifies users 98% of the time).
- P(B | not A) = 1 - 0.95 = 0.05 (the test correctly identifies nonusers 95% of the time).
First, we need to calculate P(B), the probability of Jim's test coming back positive. We can do this using the law of total probability:
P(B) = P(B | A) * P(A) + P(B | not A) * P(not A).
P(not A) is the probability of Jim not being a heroin user, which is 1 - P(A). Therefore:
P(not A) = 1 - 0.06 = 0.94.
P(B) = (0.98 * 0.06) + (0.05 * 0.94) = 0.0588 + 0.047 = 0.1058.
Now we can substitute these values into Bayes' theorem to find P(A | B):
P(A | B) = (0.98 * 0.06) / 0.1058 ≈ 0.0558 / 0.1058 ≈ 0.527.
Therefore, the probability that Jim is a heroin user given that his test came back positive is approximately 0.527, or 52.7%.