Use the sample data in the table.

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 3 of the 98 test subjects are randomly selected without replacement, find the probability that they all had false positive results.

To find the probability that all three of the selected test subjects had false positive results, we can proceed with the following steps:

### Step 1: Determine the total number of test subjects and the number of false positives.

From the table:

- Total number of test subjects = \(15 + 42 + 32 + 9 = 98\)
- Number of false positive results = 15

### Step 2: Calculate the probability of selecting 3 false positives without replacement.

When selecting without replacement, the probability of each selection depends on the previous one. The probability that all three selected subjects had false positive results is calculated as follows:

1. **First selection:** Probability of choosing a false positive = \(\frac{15}{98}\)
2. **Second selection:** After choosing one false positive, there are 14 false positives left out of 97 subjects. So, the probability = \(\frac{14}{97}\)
3. **Third selection:** After choosing two false positives, there are 13 false positives left out of 96 subjects. So, the probability = \(\frac{13}{96}\)

### Step 3: Multiply the probabilities together.

\[
P(\text{3 false positives}) = \frac{15}{98} \times \frac{14}{97} \times \frac{13}{96}
\]

Let's calculate this:

\[
P(\text{3 false positives}) = \frac{15 \times 14 \times 13}{98 \times 97 \times 96}
\]

Now, let's compute this:

\[
P(\text{3 false positives}) = \frac{2730}{912576} \approx 0.002992
\]

### Final Answer:

The probability that all three selected subjects had false positive results is approximately **0.00299**.

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Would you like more details or have any questions? Here are some related questions:

1. What is the probability of selecting one false positive and two true positives?
2. How does the probability change if the selections are made with replacement?
3. What is the probability of selecting one false positive and two subjects who did not lie?
4. How does the sample size impact the probability calculation?
5. What are the implications of a false positive in this context?

**Tip:** When dealing with probability without replacement, remember that the total number of subjects reduces with each selection.

Calculate the probability of selecting three test subjects who all had false positive results in polygraph tests. This problem involves understanding probability and combinatorics with a specific application to polygraph test results.

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