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.