Question

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 \({ }^{\prime} \mathrm{No}\)," 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?

Short answer

The probability is approximately 81.43\%.

Step-by-step solution

Define Events and Probabilities

Let's define some probabilities based on the information given:- Let \( T \) be the event that an applicant is trustworthy, which has a probability of \( P(T) = 0.95 \).- The event \( \bar{T} \) (not trustworthy) thus has a probability of \( P(\bar{T}) = 1 - 0.95 = 0.05 \).- Let \( L \) be the event that a lie is detected.- According to the problem, - \( P(L|T) = 0.15 \), the probability of a true statement being incorrectly identified as a lie (false positive). - \( P(L|\bar{T}) = 0.65 \), the probability of a lie being correctly identified.

Use Bayes' Theorem

To find the probability that an applicant is trustworthy given that they were detected as lying, we use Bayes' Theorem:\[ P(T|L) = \frac{P(L|T) \cdot P(T)}{P(L)} \]We need to calculate \( P(L) \), the total probability of detecting a lie.

Calculate Total Probability of Detecting a Lie

The total probability \( P(L) \) is given by the law of total probability:\[ P(L) = P(L|T) \cdot P(T) + P(L|\bar{T}) \cdot P(\bar{T}) \]Plug in the numbers:\[ P(L) = 0.15 \cdot 0.95 + 0.65 \cdot 0.05 \]\[ P(L) = 0.1425 + 0.0325 \]\[ P(L) = 0.175 \]

Compute the Desired Probability

Substitute \( P(T) \), \( P(L|T) \), and \( P(L) \) into Bayes' theorem:\[ P(T|L) = \frac{0.15 \cdot 0.95}{0.175} \]\[ P(T|L) = \frac{0.1425}{0.175} \]\[ P(T|L) \approx 0.8143 \]Therefore, the probability that a job applicant rejected under suspicion of dishonesty was actually trustworthy is approximately 81.43\%.

Key concepts

Probability

Probability is a way of measuring how likely an event is to occur. In simple terms, it's about expressing the chance of something happening using numbers. This concept is crucial in many fields, including statistics, finance, and everyday decision-making.
When considering a lie detector test, probability helps us understand how reliable the test results might be. For example, if we say there's a 65% probability a lie will be detected, it means that out of 100 lies, around 65 might be caught by the detector. These numbers help us set expectations and make informed decisions based on the likelihood of various outcomes.
In the context of our exercise, probabilities for different events are given:

• The probability of an applicant being trustworthy (\( P(T) = 0.95 \)).
• The probability of someone not being trustworthy (\( P(\bar{T}) = 0.05 \)).
• The probability of a true statement being incorrectly identified as a lie (called a false positive) is \( P(L|T) = 0.15 \).
• The probability of a lie being identified correctly is \( P(L|\bar{T}) = 0.65 \).

Understanding these probabilities is essential for applying Bayes' Theorem and making predictions about potential outcomes. These predictions are crucial in making decisions based on the reliability of the polygraph test.

Lie Detector Tests

Lie detector tests, commonly known as polygraphs, are tools used to detect deception by measuring physiological responses such as heart rate, blood pressure, and respiration. Despite their controversial nature, these tests still play a role in various contexts, such as pre-employment screenings. However, their accuracy is often debated.
The key idea behind a polygraph test is that changes in physiological responses can indicate when a person is lying. However, these changes can also occur for reasons unrelated to deception, making it a challenge to determine whether someone is actually lying. In the exercise example, the test can detect 65% of lies but also falsely identifies 15% of truthful statements as lies. This scenario highlights the challenges of using lie detectors in crucial decision-making processes.
Understanding these aspects of lie detector tests is vital, especially considering their potential for producing false positives. Employers should weigh the benefits and risks of using such tests, given that they might not always provide accurate assessments of honesty.

False Positives

A false positive occurs when a test incorrectly identifies a condition or result that is not present. In the case of lie detectors, this means identifying a truthful statement as a lie. False positives are significant in analyzing the reliability and effectiveness of any test.For the polygraph test exercise, the false positive rate is given as 15% (\( P(L|T) = 0.15 \)). This means that, despite answering truthfully, some job applicants are wrongly marked as liars. This can have serious repercussions, such as undeserved job rejections.
To understand the importance of false positives in decision-making, it's critical to consider their impact in terms of trust and fairness. They can lead to incorrect conclusions and decisions based on misleading information. Therefore, when relying on tests like polygraphs, it's essential to understand their limitations, including how they might mistakenly raise doubts about an honest individual's integrity. Such insights can guide businesses in making more balanced and informed hiring decisions.