**Use the table to answer the question.**

Results from Experiments with Polygraph Instruments Did the Subject Actually Lie?
 
No
(Did Not Lie)

Yes
(Lied)

Positive test result
(Polygraph test indicated that the subject lied.)

15
(false positive)

42
(true positive)

Negative test result 
(Polygraph test indicated that the subject did not lie.)

32
(true negative)

9
(false negative)


If one of the test subjects is randomly selected, find the probability that the subject had a positive test result or did not lie.

Answer 1 Question 4

To find the probability that the subject had a positive test result or did not lie, we need to use the principle of probability for the union of two events.

Let:
- \( A \) be the event that the subject had a positive test result.
- \( B \) be the event that the subject did not lie.

We need to find \( P(A \cup B) \), which is given by:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

### Step 1: Calculate \( P(A) \)
The probability of a positive test result \( P(A) \) is:

\[
P(A) = \frac{\text{Number of positive test results}}{\text{Total number of test subjects}}
\]

Positive test results include both true positives and false positives:

\[
P(A) = \frac{42 + 15}{15 + 42 + 32 + 9} = \frac{57}{98}
\]

### Step 2: Calculate \( P(B) \)
The probability that the subject did not lie \( P(B) \) is:

\[
P(B) = \frac{\text{Number of subjects who did not lie}}{\text{Total number of test subjects}}
\]

Subjects who did not lie include both true negatives and false positives:

\[
P(B) = \frac{32 + 15}{98} = \frac{47}{98}
\]

### Step 3: Calculate \( P(A \cap B) \)
The probability that the subject had a positive test result **and** did not lie \( P(A \cap B) \) is:

\[
P(A \cap B) = \frac{\text{Number of false positives}}{\text{Total number of test subjects}} = \frac{15}{98}
\]

### Step 4: Calculate \( P(A \cup B) \)

Now, we can calculate \( P(A \cup B) \):

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{57}{98} + \frac{47}{98} - \frac{15}{98} = \frac{89}{98}
\]

So, the probability that the subject had a positive test result or did not lie is \( \frac{89}{98} \) or approximately 0.908.

Would you like more details or have any other questions?

Here are 5 related questions for further practice:
1. What is the probability that the subject lied or had a negative test result?
2. What is the probability that the subject had a negative test result or did not lie?
3. What is the probability that the subject did not lie and had a negative test result?
4. What is the probability that the subject had a false test result (either false positive or false negative)?
5. What is the probability that the subject had a true test result (either true positive or true negative)?

**Tip:** Always make sure to correctly identify the intersection of events when using the union formula in probability problems.

Learn how to calculate the probability of a positive test result or that the subject did not lie using probability theory. This problem involves finding the union of two events: positive test result and no lie, with detailed step-by-step calculations and explanations.

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