Final answer:
Using Bayes' theorem, the posterior probability that an athlete is a drug user given that her test result is negative is calculated to be approximately 0.437%.
Step-by-step explanation:
The question is asking about the posterior probability that an athlete is a drug user given that her drug test result is negative.
To solve this, we use Bayes' theorem, which takes into account the prior probability of being a drug user, the probability of a negative test result for both drug users and nonusers, and normalizes the likelihood by considering the overall probability of a negative result.
Let's define the probabilities:
- The prior probability of being a drug user (P(User)) is 0.05.
- The probability of a negative result given that the person is a drug user (P(Negative|User)) is 0.06.
- The probability of a negative result given that the person is a nonuser (P(Negative|Nonuser)) is 0.72.
- The prior probability of being a nonuser (P(Nonuser)) is 1 - P(User) = 0.95.
The overall probability of a negative result (P(Negative)) is calculated as:
P(Negative) = P(Negative|User) × P(User) + P(Negative|Nonuser) × P(Nonuser)
P(Negative) = 0.06 × 0.05 + 0.72 × 0.95
P(Negative) = 0.003 + 0.684
P(Negative) = 0.687
We then calculate the posterior probability that someone is a drug user given a negative test result (P(User|Negative)) as follows:
P(User|Negative) = [P(Negative|User) × P(User)] / P(Negative)
P(User|Negative) = [0.06 × 0.05] / 0.687
P(User|Negative) = 0.003 / 0.687
P(User|Negative) ≈ 0.00437 or 0.437%
Therefore, the posterior probability that an athlete is a drug user given a negative test result is approximately 0.437%.