91Ó°ÊÓ

Lie detectors are controversial instruments, barred from use as evidence in many courts. Nonetheless, many employers use lie detector screening as part of their hiring process in the hope that they can avoid hiring people who might be dishonest. There has been some research, but no agreement, about the reliability of polygraph tests. Based on this research, suppose that a polygraph can detect \(65 \%\) of lies, but incorrectly identifies \(15 \%\) of true statements as lies. A certain company believes that \(95 \%\) of its job applicants are trustworthy. The company gives everyone a polygraph test, asking, "Have you ever stolen anything from your place of work?" Naturally, all the applicants answer "No," but the polygraph identifies some of those answers as lies, making the person ineligible for a job. What's the probability that a job applicant rejected under suspicion of dishonesty was actually trustworthy?

Step by step solution

01

Define the Probabilities

We define the following probabilities: let \( T \) be the event that an applicant is trustworthy and \( L \) be the event that a lie is detected. We are interested in finding \( P(T|L) \), the probability that an applicant is trustworthy given that a lie is detected. From the problem statement, we know:\( P(L|T^c) = 0.65 \) (polygraph detects lies correctly) \( P(L|T) = 0.15 \) (polygraph incorrectly identifies true statements as lies) \( P(T) = 0.95 \) (applicant being trustworthy), thus \( P(T^c) = 0.05 \).

02

Apply Bayes' Theorem

We use Bayes' theorem to find the desired probability:\[ P(T|L) = \frac{P(L|T) \cdot P(T)}{P(L)}. \] We need to calculate \( P(L) \), the total probability that a lie is detected regardless of whether the applicant is trustworthy or not.

03

Calculate the Total Probability of a Lie Being Detected

To find \( P(L) \), we calculate:\[ P(L) = P(L|T) \cdot P(T) + P(L|T^c) \cdot P(T^c) = 0.15 \times 0.95 + 0.65 \times 0.05 = 0.1425 + 0.0325 = 0.175. \]

04

Substitute Values into Bayes' Theorem

Now substitute the values into Bayes' theorem:\[ P(T|L) = \frac{0.15 \times 0.95}{0.175}. \]

05

Final Calculation

Calculate \( P(T|L) \) using the values obtained:\[ P(T|L) = \frac{0.1425}{0.175} = 0.8143. \] Thus, the probability that a job applicant rejected under suspicion of dishonesty was actually trustworthy is approximately \( 0.8143 \), or 81.43%.

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Key Concepts

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

Understanding Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already happened. It's a foundational concept in probability, particularly when you're dealing with dependent events. In this exercise, we are interested in the probability that an applicant is trustworthy given that a lie is detected \( P(T|L) \). This is different from simply considering the probability of being trustworthy or the probability of a lie being detected independently.
To calculate conditional probabilities, we often use **Bayes' Theorem**. Bayes' Theorem helps us reverse conditional probabilities and find the relationship between different events. It's especially useful in situations like this exercise where we want to know the probability of the underlying truth given some observed evidence, which is crucial for decision-making in fields like diagnostics and risk assessment.

The Role of Polygraph Tests

Polygraph tests, often called lie detectors, are tools used to determine if someone is being truthful. They measure physiological responses such as heart rate, blood pressure, and breathing patterns while the subject answers questions. The idea is that certain physiological responses can indicate stress or deception. However, the accuracy and reliability of these tests are often debated.
In our example, it is given that polygraph tests detect 65% of the lies, but they have a downside—they incorrectly label 15% of truthful statements as lies. This limitation highlights a significant issue with polygraphs: they can produce **false positives**. These inaccuracies make it crucial to understand the broader context when reading polygraph results. Polygraph results should rarely be the sole basis for decisions like employment, due to potential errors.

The Basics of Probability Theory

Probability theory is the branch of mathematics concerned with the analysis of random phenomena. It provides the tools to deal with uncertainty and model a scenario like the one in this exercise. Basic probability measures how likely an event is to occur and can be expressed in fractions or percentages.
- Probability of an event happening (e.g., being truthful) is denoted as \( P(T) \).
- The probability of the opposite event occurring is \( P(T^c) \) which is \( 1 - P(T) \).
These foundational principles allow us to calculate both independent events and more complex scenarios involving conditional probabilities, such as determining an individual's trustworthiness given the test results.

Navigating False Positives

False positives occur when a test indicates that a condition or attribute is present when it is not. In the context of the polygraph test, a false positive happens when an applicant who is actually trustworthy is identified as lying. This type of error is not uncommon in various testing scenarios, from medical tests to security screenings.
False positives are critical to address because they can lead to incorrect conclusions and inappropriate actions, such as unfairly disqualifying a truthful job candidate. Understanding the rate of false positives \( P(L|T) \) is essential in interpreting the test results correctly and making informed decisions. In this exercise, the goal was to find out how likely it is that someone flagged as lying is, in reality, honest, highlighting the importance of questioning and verifying test results before taking action.