Question

In testing prospective employees for drug use, companies need to remember that the tests are not \(100 \%\) reliable. Suppose a company uses a test that is \(98 \%\) accurate - that is, it correctly identifies a person as a drug user or nonuser with probability \(.98-\) and to reduce the chance of error, each job applicant must take two tests. Assume that the outcomes of the two tests on the same person are independent events, and find the following probabilities: a. A nonuser fails both tests. b. A drug user is detected (i.e., he or she fails at least one test). c. A drug user passes both tests.

Short answer

a. A nonuser fails both tests. b. A drug user is detected (i.e., they fail at least one test). c. A drug user passes both tests. Answer: a. The probability of a nonuser failing both tests is 0.0004. b. The probability of a drug user being detected is 0.0396. c. The probability of a drug user passing both tests is 0.9604.

Step-by-step solution

Calculate the probabilities for a single test

First, we need to determine the probabilities for passing and failing a single test for drug users and nonusers - P(passing a test | nonuser) = \(0.98\) - P(failing a test | nonuser) = \(1 - 0.98 = 0.02\) - P(passing a test | drug user) = \(0.98\) - P(failing a test | drug user) = \(0.02\)

Calculate the probability of a nonuser failing both tests

Since the outcomes of the two tests are independent events, we can use the multiplication rule to solve for this probability. P(failing both tests | nonuser) = P(failing a test | nonuser) * P(failing a test | nonuser) = \(0.02 * 0.02 = 0.0004\)

Calculate the probability of a drug user passing both tests

We can use a similar approach for calculating the probability of a drug user passing both tests. P(passing both tests | drug user) = P(passing a test | drug user) * P(passing a test | drug user) = \(0.98 * 0.98 = 0.9604\)

Calculate the probability of a drug user being detected

A drug user is detected when they fail at least one test. To do this, we can use the complement rule and subtract the probability of a drug user passing both tests from \(1\). P(drug user is detected) = 1 - P(passing both tests | drug user) = 1 - 0.9604 = 0.0396 The following probabilities are obtained: a. A nonuser fails both tests: \(0.0004\) b. A drug user is detected: \(0.0396\) c. A drug user passes both tests: \(0.9604\)

Key concepts

Independent Events

In probability, understanding independent events is crucial. Two events are independent if the occurrence of one does not affect the occurrence of the other. This means that whether or not Event A occurs, it has absolutely no impact on the probability of Event B occurring.
Consider drug tests as an example. Each test given to an individual is treated as an independent event. This implies that the result of one test does not influence the result of another test. This characteristic allows us to calculate the combined probability of two events happening together by simply multiplying their individual probabilities.

• For instance, if a nonuser fails a single test with a probability of 0.02, and since each test is independent, the probability of failing two independent tests is found by multiplying: 0.02 * 0.02 = 0.0004.

This simple multiplication is the foundation of dealing with independent events and it's key to solving problems involving multiple independent occurrences.

Complement Rule

The Complement Rule is a very useful tool in probability, especially when it is easier to calculate the probability of the complementary event. The complement of an event is the scenario where the event does not occur.
In probability terms, if Event A is to happen, then the probability of Event A not happening is given by the complement rule as 1 minus the probability of A:

• The formula is: P(A') = 1 - P(A).

In our drug test scenario, a drug user being detected means they fail at least one test. It's often simpler to calculate the probability of the complementary event, which is them passing both tests, and subtracting that from 1.
Thus, we have: P(drug user is detected) = 1 - P(passing both tests). This principle helps streamline many calculations in probability.

Multiplication Rule

The Multiplication Rule is pivotal whenever you're dealing with the probability of two independent events occurring together. If you know the probabilities of two events occurring separately and want to find the probability that they'll occur at the same time, the multiplication rule comes into play.
For independent events, the rule is straightforward: The probability of both events happening is simply the product of their individual probabilities. In mathematical terms, if A and B are independent events, then

• P(A and B) = P(A) * P(B).

For instance, knowing that each drug test's outcome is independent, we can compute the probability of an applicant failing both tests by multiplying the probability of failing one test by itself, as shown in the solution to our exercise above.

False Positives

A false positive in testing occurs when the test incorrectly indicates the presence of a condition, when it is not actually present. This is a crucial concept in drug testing because misclassification can have significant consequences, such as denying employment without proper cause.
In the context of our problem, a false positive would happen if a nonuser of drugs fails the test. Understanding this risk is vital for interpreting test results accurately and planning how additional tests could mitigate it.

• The tests mentioned are 98% accurate, meaning there's a 2% chance of a false positive for any given test.

By requiring applicants to take two independent tests, companies aim to reduce the likelihood of false positives, capitalizing on the multiplication rule to further lower the probability of misclassification.