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Ch. 4 - Probability
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 4 - ProbabilityProblem 4.3.18
Chapter 4, Problem 4.3.18

In Exercises 17–20, refer to the accompanying table showing results from experiments conducted by researchers Charles R. Honts (Boise State University) and Gordon H. Barland (Department of Defense Polygraph Institute). In each case, it was known whether or not the subject lied, so the table indicates when the polygraph (lie detector) test was correct.


"Table showing polygraph results: 15 true negatives, 42 false positives, 32 false negatives, 9 true positives."


False Negative Find the probability of selecting a subject with a negative polygraph result, given that the subject lied. What would be an unfavorable consequence of this error?

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Step 1: Understand the problem. We are tasked with finding the probability of selecting a subject with a negative polygraph result (polygraph indicated the subject did not lie), given that the subject actually lied. This is a conditional probability problem.
Step 2: Identify the relevant data from the table. From the table, the number of subjects who lied and had a negative polygraph result is 9. The total number of subjects who lied is the sum of the values in the 'Subject Lied' column, which is 42 + 9 = 51.
Step 3: Write the formula for conditional probability. The probability of event A given event B is calculated as: P(A|B) = P(A ∩ B) / P(B). Here, A is 'negative polygraph result' and B is 'subject lied'.
Step 4: Substitute the values into the formula. P(A ∩ B) is the number of subjects who lied and had a negative polygraph result (9), divided by the total number of subjects (15 + 42 + 32 + 9 = 98). P(B) is the number of subjects who lied (51), divided by the total number of subjects (98).
Step 5: Discuss the unfavorable consequence of this error. A false negative means the polygraph test incorrectly indicates that a lying subject did not lie. This could lead to a situation where a guilty person is wrongly deemed truthful, potentially undermining justice or security.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

False Negative

A false negative occurs when a test fails to identify a condition that is present. In the context of the polygraph test, it means that the test indicated a subject did not lie when, in fact, they did. This type of error can lead to significant consequences, such as allowing a deceptive individual to go unpunished.
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Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this scenario, calculating the probability of selecting a subject with a negative polygraph result given that they lied involves using the relevant counts from the table to determine the ratio of false negatives to the total number of subjects who lied.
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Confusion Matrix

A confusion matrix is a table used to evaluate the performance of a classification model by comparing predicted classifications to actual outcomes. In this case, the matrix helps visualize the results of the polygraph tests, showing true positives, false positives, true negatives, and false negatives, which are essential for understanding the accuracy and reliability of the test.
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Related Practice
Textbook Question

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Textbook Question

Notation For a polygraph (lie detector) used when a subject is presented with a question, let L= the subject lied and let Y = the polygraph indicated that the subject told a lie. Use your own words to translate the notation P (Y|L) into a verbal statement.

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Textbook Question

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