Chapter 7, Problem 14CRE

Refer to the previous exercise. Suppose that because of the high rate of false-positives for the drug test Mumble.com has instituted a mandatory independent second test for those who test positive on the first test.

1. a. If one employee is selected at random, what is the probability that the selected employee uses drugs and tests positive twice?
2. b. If one employee is selected at random, what is the probability that the employee tests positive twice?
3. c. If we know that the randomly chosen employee has tested positive twice, what is the probability that he or she uses drugs?
4. d. What is the chance that an individual who does use drags doesn’t test positive twice (either this employee tests negative on the first round and doesn’t need a retest, or this employee tests positive the first time and that result is followed by a negative result on the retest)?
5. e. Discuss the benefits and drawbacks of using a retest scheme such as the one proposed in this question.

CR7.13    Suppose that a new Internet company Mumble.com requires all employees to take a drug test. Mumble.com can afford only the inexpensive drug test—the one with a 5% false-positive rate and a 10% false-negative rate. (That means that 5% of those who are not using drugs will incorrectly test positive and that 10% of those who are actually using drugs will test negative.) Suppose that 10% of those who work for Mumble.com are using the drugs for which Mumble is checking. (Hint: It may be helpful to draw a tree diagram to answer the questions that follow.)

a. If one employee is chosen at random, what is the probability that the employee both uses drugs and tests positive?

b. If one employee is chosen at random, what is the probability that the employee does not use drugs but tests positive anyway?

c. If one employee is chosen at random, what is the probability that the employee tests positive?

d. If we know that a randomly chosen employee has tested positive, what is the probability that he or she uses drugs?