Let's break down the problem step by step to calculate the probability and evaluate whether the event is unusual.

Data Summary

The table provides the following counts for the polygraph results:

Did the Subject Actually Lie?	Positive test result (Polygraph indicated lie)	Negative test result (Polygraph indicated no lie)
No (Did not lie)	15 (False positive)	32 (True negative)
Yes (Lied)	42 (True positive)	9 (False negative)

Total Number of Subjects:

• Total subjects: $15 + 42 + 32 + 9 = 98$

Probability Calculation:

We need to find the probability that in each of the four randomly selected subjects, the polygraph indicates that the subject lied.

There are two groups for which the polygraph test indicates that the subject lied:

• False positives (15 subjects): The test wrongly indicated a lie when the subject did not lie.
• True positives (42 subjects): The test correctly indicated a lie when the subject actually lied.

Thus, the total number of subjects where the polygraph indicated a lie is: $15 \, (\text{false positive}) + 42 \, (\text{true positive}) = 57 \, (\text{subjects indicated as lying})$

Now, we are selecting 4 subjects randomly. The probability that each selected subject has a positive test result (i.e., the polygraph indicated they lied) is computed using the formula for probability without replacement:

$P(\text{All 4 subjects lied}) = \frac{\text{Number of subjects indicating a lie in the first trial}}{\text{Total number of subjects}} \times \frac{\text{Number of subjects indicating a lie in the second trial}}{\text{Total number of subjects} - 1} \times \cdots$

This probability is:

$P = \frac{57}{98} \times \frac{56}{97} \times \frac{55}{96} \times \frac{54}{95}$

Now, let's calculate the exact value.

Calculating the Probability:

$P = \frac{57}{98} \times \frac{56}{97} \times \frac{55}{96} \times \frac{54}{95} \approx 0.375$

So, the probability that, in each case, the polygraph indicated that the subject lied is approximately 0.375, or 37.5%.

Is this event unusual?

• A typical threshold for an event to be considered "unusual" is a probability of less than 5%. Since the probability here is much greater than 5% (it's 37.5%), the event is not unusual.

Conclusion:

• The probability is 37.5%, which is greater than 5%.
• Hence, the event is not unusual.

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Further Exploration:

1. How would the probability change if we selected 5 subjects instead of 4?
2. How does the total number of subjects in the experiment affect the probability?
3. How can this probability be calculated using the combination formula for sampling without replacement?
4. What factors could contribute to a polygraph test's accuracy in predicting whether someone is lying?
5. What is the interpretation of false positive and false negative results in a polygraph test?

Tip:

When calculating probabilities without replacement, always adjust the denominator in each successive fraction to account for the reduced total number of items remaining.