91Ó°ÊÓ

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?

Step by step solution

01

Understanding the Problem

We are given: The probability that an employee uses drugs (P(D)) = 0.10, The probability that an employee doesn’t use drugs (P(ND)) = 1 – P(D) = 0.90, The probability that the test is positive given the person is a drug user (P(T|D)) = 1 – false negative rate = 1 - 0.10 = 0.90, The probability that the test is positive given the person is not a drug user (P(T|ND)) = false positive rate = 0.05.

02

Probability of a Drug User Testing Positive

To find P(D and T), we multiply P(D) and P(T|D), thus P(D and T) = P(D) * P(T|D) = (0.10) * (0.90) = 0.09.

03

Probability of a Non-Drug User Testing Positive

To find P(ND and T), we multiply P(ND) and P(T|ND), thus P(ND and T) = P(ND) * P(T|ND) = (0.90) * (0.05) = 0.045.

04

Probability that any Employee will Test Positive

To find this, we add all the ways an employee can test positive. Thus, P(T) = P(D and T) + P(ND and T) = 0.09 + 0.045 = 0.135.

05

Probability that an Employee Uses Drugs Given a Positive Test

Use the formula for conditional probability P(D|T) = P(D and T) / P(T) = 0.09 / 0.135 = 0.6667.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

False Positive Rate

The concept of a false positive rate is crucial, especially in scenarios where tests are administered for rare conditions or substances, like drugs in the workplace. A false positive occurs when a test incorrectly indicates the presence of a substance or condition, even though it is not present. For example, in our exercise with Mumble.com, the drug test has a 5% false positive rate. This implies that 5% of employees who do not use drugs will still test positive.

Understanding the false positive rate is important for determining how reliable a test is. A test with a high false positive rate could lead to unnecessary distress for employees and potentially misinformed decisions by employers. It reminds us to consider the broader implications of testing protocols and the importance of additional verification.

False Negative Rate

A false negative rate is the opposite of a false positive rate. It refers to the percentage of tests where the outcome is negative though the condition or substance is present. In the context of Mumble.com's drug test, the false negative rate is 10%. This means that 10% of employees who do use drugs will receive a negative test result.

The significance of the false negative rate lies in its potential to overlook actual cases. Resulting in drug users not being identified in the drug testing process. Balancing false positives and false negatives is vital for any testing strategy, as each has different consequences. Weighing these rates can help decision-makers choose the most appropriate testing methods for their needs.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It's a fundamental concept in probability that helps us understand the likelihood of complex events. For instance, in the Mumble.com situation, we want to find the probability that an employee uses drugs given that they tested positive.

To calculate this, we use the formula:
\[ P(D|T) = \frac{P(D \text{ and } T)}{P(T)} \]
Where:

• \( P(D|T) \) is the conditional probability of using drugs given a positive test.
• \( P(D \text{ and } T) \) is the probability of using drugs and testing positive (which we've found to be 0.09).
• \( P(T) \) is the total probability of testing positive from either drug users or non-users (0.135 from step 4 of our solution).

Plugging in the values, we find that \( P(D|T) = \frac{0.09}{0.135} = 0.6667 \). Therefore, if an employee tests positive, there is approximately a 66.67% chance that they use drugs.

Tree Diagram

A tree diagram is a visual representation that helps in breaking down complex probabilities into simpler, more manageable parts. It is especially useful when dealing with multiple conditional outcomes such as those in drug testing.

Visualizing the problem using a tree diagram involves the following steps:

• Begin with the overall population of Mumble.com employees.
• Split this into two branches: those who use drugs (10%) and those who do not (90%).
• For each branch, draw further splits indicating the outcomes of the drug test: testing positive or negative.

For the drug users, this results in two branches – testing positive with a 90% probability and testing negative with a 10% probability. For the non-users, the branches are testing positive with a 5% probability and testing negative with a 95% probability.

The tree diagram provides a clear map of all possible outcomes and their associated probabilities, allowing for easy calculation of complex probabilities by following each branch. It's a powerful tool for visual learners and anyone tackling intricate probability problems.