Question

question 20 (10 points)
use bayes theorem to find the indicated probability
some employers use lie detector tests to screen job applicants. lie detector tests are
not completely reliable. suppose that in a lie detector test, 65% of lies are identified
as lies and that 14% of true statements are also identified as lies.
a company gives its job applicants a polygraph test, asking \did you tell the truth on
your job application?\. suppose that 93% of the job applicants tell the truth during
the
polygraph test.
what is the probability that a person who fails the test was actually telling the truth?

Explanation:

Step1: Define Events

Let \( T \) be the event that a person tells the truth, and \( F \) be the event that a person fails the test. We know:

• \( P(T) = 0.93 \) (probability of telling the truth)
• \( P(

eg T) = 1 - 0.93 = 0.07 \) (probability of lying)

• \( P(F|T) = 0.14 \) (probability of failing given truth)
• \( P(F|

eg T) = 0.65 \) (probability of failing given lie)

Step2: Apply Bayes' Theorem

Bayes' theorem states \( P(T|F) = \frac{P(F|T)P(T)}{P(F|T)P(T) + P(F|
eg T)P(
eg T)} \)

Step3: Calculate Denominator

First, calculate the denominator \( P(F|T)P(T) + P(F|
eg T)P(
eg T) \):
\( (0.14 \times 0.93) + (0.65 \times 0.07) \)
\( = 0.1302 + 0.0455 = 0.1757 \)

Step4: Calculate Numerator

The numerator is \( P(F|T)P(T) = 0.14 \times 0.93 = 0.1302 \)

Step5: Compute \( P(T|F) \)

\( P(T|F) = \frac{0.1302}{0.1757} \approx 0.7411 \)

Answer:

\( \approx 0.741 \) (or 74.1% when expressed as a percentage)