It is known that 10 % of a school's athlete population uses drugs. The Athletic Director gives a drug test which registers correctly as positive in 82 % of the cases where the drug is present and correctly as negative in 96 % of the cases where the drug is absent. One athlete is randomly selected and tested. Find the probability that the athlete is NOT a drug user, given that the test result is negative. Give your answer as a decimal number, with 4 decimal places.

Question

It is known that 10 % of a school's athlete population uses drugs. The Athletic Director gives a drug test which registers correctly as positive in 82 % of the cases where the drug is present and correctly as negative in 96 % of the cases where the drug is absent. One athlete is randomly selected and tested. Find the probability that the athlete is NOT a drug user, given that the test result is negative. Give your answer as a decimal number, with 4 decimal places.

Answer 1

To find the probability that the athlete is NOT a drug user given that the test result is negative, we can use Bayes' theorem. Bayes' theorem states that the conditional probability of an event A given an event B can be calculated as:

P(A | B) = (P(B | A) * P(A)) / P(B)

In this case, we want to find the probability that the athlete is NOT a drug user (A) given that the test result is negative (B). Let's break down the information we have:

P(A) = probability that the athlete is NOT a drug user = 1 - 0.10 = 0.90 (since we know that 10% of the athlete population uses drugs)
P(B | A) = probability that the test result is negative given that the athlete is NOT a drug user = 0.96 (given in the problem)
P(B) = probability that the test result is negative

To find P(B), we need to consider both the cases where the athlete is a drug user and where the athlete is not a drug user.

Case 1: The athlete is a drug user
P(B | A') = probability that the test result is negative given that the athlete is a drug user = 1 - 0.82 (since a positive test result registers as 1 - 0.82 = 0.18)
P(A') = probability that the athlete is a drug user = 0.10 (given in the problem)

Case 2: The athlete is not a drug user
P(B | A) = probability that the test result is negative given that the athlete is not a drug user = 0.96 (given in the problem)
P(A) = probability that the athlete is not a drug user = 0.90 (calculated earlier)

Now, let's calculate P(B):

P(B) = P(B | A) * P(A) + P(B | A') * P(A')
= 0.96 * 0.90 + 0.18 * 0.10
= 0.864 + 0.018
= 0.882

Finally, we can calculate P(A | B) using Bayes' theorem:

P(A | B) = (P(B | A) * P(A)) / P(B)
= (0.96 * 0.90) / 0.882
= 0.864 / 0.882
≈ 0.9796

Therefore, the probability that the athlete is NOT a drug user, given that the test result is negative, is approximately 0.9796 (rounded to 4 decimal places).