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The reliability of polygraph tests, commonly known as lie detector tests, is a topic of much debate. In the exercise, we deal with the probability that trustworthy individuals will pass such a test, when false positives—a scenario where the test fails a truthful person—occur 15% of the time.

Calculating the probability for one individual to pass the test is straightforward. With an 85% chance to pass (100% - 15% failure rate), for one individual, we get a probability of 0.85. When considering multiple individuals, we must account for the fact that each test is independent, meaning one person's result doesn't affect another's. To find the probability that all 10 pass, we would need to multiply the probability of one person passing the test by itself 10 times, which can be expressed mathematically as \(0.85^{10}\) and amounts to approximately 19.68%. This illustrates the compound effect of independent probabilities in binomial distributions.

When dealing with binomial distributions, two important statistical measures are the mean and the standard deviation. These values help us to understand the expected value and the variability of the random variable under consideration. In our case, the random variable \(X\) is the number of individuals who fail the polygraph test.

The mean of a binomial distribution is calculated as \(\mu = n\cdot p\), where \(n\) is the number of trials and \(p\) is the probability of success on a single trial. The standard deviation gives us an idea of how much variation there is from the mean and is calculated using the formula \(\sigma = \sqrt{n\cdot p\cdot (1-p)}\).

For the exercise involving 500 FBI agents, applying these formulas gives us a mean of 75 and a standard deviation of approximately 7.423. This means that, on average, we can expect 75 failures, but due to natural variability in the test results, the actual number may typically vary by about 7.423 failures from this average.

Z-scores are a standard tool for understanding how a single data point compares with a standard distribution. A z-score represents the number of standard deviations a point is from the mean. In a normal distribution context, it can tell us how unusual or common a particular outcome is.

To calculate the z-score, we use the formula \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the value we are comparing to the mean (\(\mu\)) of the data, and \(\sigma\) is the standard deviation. In the FBI agents example, when we calculate a z-score for 25 failed tests out of 500, we get a z-score of approximately -6.73, which indicates that the result is 6.73 standard deviations below the mean.

Typically, a z-score beyond -3 or +3 is considered very unusual, as it falls outside the range where we expect 99.7% of all data points to lie in a normal distribution. Hence, getting a result of fewer than 25 failures is not just surprising; it is extraordinarily rare given the expected distribution, suggesting factors outside of randomness or questioning the initial assumption that all agents were, in fact, trustworthy.