To reduce theft, the Meredeth Company screens all its employees with a lie detector test that is known to be correct 90 percent of the time (for both guilty and innocent subjects). George Meredeth decides to fire all employees who fail the test. Suppose 5 percent of the employees are guilty of theft.
a. What proportion of the workers are fired?
b. Of the workers fired, what proportion are actually guilty?
c. Of the workers not fired, what proportion are guilty?
d. What do you think of George’s policy?
To find the answers to these questions, we can use conditional probability and the given information.
a. To determine the proportion of workers fired, we need to calculate the probability of failing the lie detector test.
Let's assume there are 100 employees in total. Since 5 percent of the employees are guilty of theft, that means there are 5 guilty employees and 95 innocent employees.
The lie detector is known to be correct 90 percent of the time, so it means it has a 90 percent accuracy rate for both guilty and innocent subjects.
Out of the 5 guilty employees, the lie detector test will correctly identify 90 percent of them as guilty, which is 4.5 employees. It will falsely identify the remaining 0.5 employee as innocent.
Out of the 95 innocent employees, the test will falsely identify 10 percent of them as guilty, which is 9.5 employees. The remaining 85.5 employees will be correctly identified as innocent.
The total number of employees who fail the test is the sum of falsely identified guilty employees and those who were actually guilty: 0.5 + 9.5 = 10 employees.
Therefore, the proportion of workers fired is 10/100 = 0.1 or 10%.
b. To determine the proportion of workers fired who are actually guilty, we need to consider the 10 employees who failed the test.
Out of the 10 employees who failed the test, only 0.5 of them were actually guilty. Therefore, the proportion of workers fired who are actually guilty is 0.5/10 = 0.05 or 5%.
c. To find the proportion of workers not fired who are guilty, we need to consider the innocent employees correctly identified by the test.
Out of the 85.5 innocent employees, none of them were falsely identified as guilty. Therefore, the proportion of workers not fired who are guilty is 0/85.5 = 0%.
d. George's policy has resulted in 10% of the employees being fired, but only 5% of them are actually guilty. This means that 95% of the employees fired by the lie detector test are innocent. Additionally, the test falsely identifies 10% of innocent employees as guilty. These results suggest that George's policy may not be effective and may lead to unfair consequences for innocent employees.