Problem 63 - College Algebra

Question

The president of a large company with \(10,000\) employees is considering mandatory cocaine testing for every employee. The test that would be used is \(90 \%\) accurate, meaning that it will detect \(90 \%\) of the cocaine users who are tested, and that \(90 \%\) of the nonusers will test negative. This also means that the test gives \(10 \%\) false positive. Suppose that \(1 \%\) of the employees actually use cocaine. Find the probability that someone who tests positive for cocaine use is, indeed, a user.

Step-by-Step Solution

Answer

The probability that someone who tests positive for cocaine use is indeed a user is approximately \(8.33\%\).

Step 1: Identify the Necessary Information

Let's denote the event that an employee is a cocaine user as A, and the event that an employee tests positive as B. We want to find the conditional probability \(P(A|B)\) - the probability that an employee is a cocaine user given that they tested positive. We know that \(P(A) = 0.01\) (the probability that a randomly selected employee is a cocaine user), \(P(A') = 0.99\) (the probability that a randomly selected employee is not a cocaine user), \(P(B|A) = 0.9\) (the probability that a cocaine user tests positive), and \(P(B|A') = 0.1\) (the probability that a non-user tests positive - false positive).

Step 2: Apply Bayes' theorem

Bayes' theorem states that \(P(A|B) = \frac{P(B|A)P(A)}{P(B)}\). We are given \(P(B|A)\) and \(P(A)\), but not \(P(B)\). However, we can compute P(B) by the law of total probability: \(P(B) = P(B \cap A) + P(B \ap A) = P(B|A)P(A) + P(B|A')P(A')\).

Step 3: Calculate \(P(B)\)

Substituting the known probabilities, we get: \(P(B) = (0.9)(0.01) + (0.1)(0.99) = 0.009 + 0.099 = 0.108\).

Step 4: Substitute into Bayes' Theorem

Now that we have \(P(B)\), we can substitute all known probabilities into Bayes' theorem: \(P(A|B) = \frac{(0.9)(0.01)}{0.108}\).

Step 5: Solve

Solving the equation, we get \(P(A|B) \approx 0.0833\), or about \(8.33\%\) (rounded to the second decimal place)

Key Concepts

Conditional Probability

Conditional probability is a key concept in probability theory that refers to the probability of an event happening given that another event has already occurred. In the context of our example, we are interested in finding the probability that an employee is a cocaine user, given that they have tested positive for cocaine. This is represented mathematically as \(P(A|B)\).
To compute a conditional probability, understanding the relationship between the two events is crucial. We don't just look at the probability of the first event happening on its own, but rather in light of the occurrence of the second event. This makes conditional probability especially useful in situations where there's a need to make decisions based on evidence or prior occurrences.
In our specific example, the conditional probability tells us that there's about an 8.33% chance an employee is a cocaine user if they tested positive. Calculating this involves using Bayes' Theorem, which leverages known probabilities to find the unknown probability.

False Positive

A false positive occurs when a test incorrectly indicates the presence of a condition when it is not actually present. In the context of the employee drug testing scenario, a false positive is when the test signals that an employee uses cocaine when they do not. The problem tells us this happens 10% of the time.
Understanding false positives is vital in the context of probability and testing since it helps evaluate the reliability of a test. For any test, minimizing false positives is crucial because they can lead to unnecessary stress and consequences, especially when the findings might significantly impact individuals' lives.
Incorporating false positives into our probability calculations ensures that decisions based on test results consider both test accuracy and the likelihood of these inaccuracies. In our calculation, this is represented by \(P(B|A') = 0.1\), the probability that a non-user tests positive.

Law of Total Probability

The law of total probability is a theorem used to find the probability of an event by considering all possible ways that event can occur. In our problem, it helps us determine \(P(B)\), the overall probability that an employee tests positive for cocaine.
The formula takes into account both the probability of a true positive and the probability of a false positive, ensuring a complete assessment of all situations leading to a positive test result. Mathematically, for event \(B\), it can be expressed as:

• \(P(B) = P(B \cap A) + P(B \cap A')\)
• Which simplifies to \(P(B|A)P(A) + P(B|A')P(A')\)

By applying the law of total probability, we incorporate all relevant probabilities, providing a comprehensive view of the circumstances under which our test indicates a positive result. This breakdown is essential for precise calculations in the Bayesian analysis.

Probability Theory

Probability theory is the mathematical framework for quantifying uncertain events and includes concepts like random variables, probability distributions, and events. It provides the tools needed to model and analyze situations involving uncertainty.
In our scenario, probability theory helps us understand the likelihood of various outcomes in the drug testing problem. Each probability mentioned in the problem, such as the accuracy of the test and the prevalence of drug use among employees, is an expression of probability theory. These probabilities provide key insights and guide the necessary calculations.
Moreover, probability theory encompasses the rules and theorems such as Bayes' Theorem and the law of total probability. It ensures that we can piece together different aspects of the problem to arrive at a coherent and logical conclusion. Understanding these principles is crucial for analyzing real-world problems, such as our company’s employee drug testing challenge.