Question

If an athlete is tested for a certain type of drug usage, then the test will come out either positive or negative. However, these tests are never perfect. Some athletes who are drug free test positive, and some who are drug users test negative. The former are called false positives; the latter are called false negatives. Assume that 5% of all athletes use drugs, 4% of all tests on drug-free athletes yield false positives, and 10% of all tests on drug users yield false negatives. Define the events as follows: Free: an athlete is drug free. User: an athlete is a drug user. Positive: an athlete is tested positive Negative: an athlete is tested negative. Then, we have P(User) - 0.05, P(Positive Free) = 0.04, and P(Negative User) - 0.10, What is the probability that a randomly selected athlete tests positive, P(Positive)? a. 0.083 b. 0.127 c. 0.308 d. 0.312

Answer 1

The probability that a randomly selected athlete tests positive for drug usage is calculated using the law of total probability, yielding a probability of 0.083.

Step-by-step explanation:

To calculate the probability that a randomly selected athlete tests positive (P(Positive)), we use the law of total probability. This considers both possibilities: the athlete being drug-free and being a drug user. We have:

The probability of being a drug user P(User) = 0.05,

The probability of a drug-free athlete testing positive P(Positive | Free) = 0.04,

The probability of a drug user testing positive P(Positive | User) = 1 - P(Negative | User) = 1 - 0.10 = 0.90,

The probability of being drug free P(Free) = 1 - P(User) = 0.95.

Now, we can apply these probabilities to the equation:

P(Positive) = P(Positive | Free) * P(Free) + P(Positive | User) * P(User)

Plugging in the values, we get:

P(Positive) = (0.04 * 0.95) + (0.90 * 0.05)

P(Positive) = 0.038 + 0.045

P(Positive) = 0.083

Therefore, the probability that a randomly selected athlete tests positive is 0.083.

Answer 2

positive...............