Question

The article on polygraph testing of FBI agents referenced in Exercise \(7.51\) indicated that the probability of a false-positive (a trustworthy person who nonetheless fails the test) is \(.15 .\) Let \(x\) be the number of trustworthy \(\mathrm{FBI}\) agents tested until someone fails the test. a. What is the probability distribution of \(x ?\) b. What is the probability that the first false-positive will occur when the third person is tested? c. What is the probability that fewer than four are tested before the first false-positive occurs? d. What is the probability that more than three agents are tested before the first false-positive occurs?

Short answer

a. The probability distribution is given by \(P(X = x) = (1-0.15)^{x-1}*0.15\), b. The probability that first false-positive will occur when the third person is tested is 0.108375, c. The probability that fewer than four are tested before the first false-positive occurs is 0.35955, d. The probability that more than three agents are tested before the first false-positive occurs is 0.64045.

Step-by-step solution

Calculate the Probability Distribution

Following the geometric distribution with parameter \(p = 0.15\) as the polygraph gives a false positive with a probability of \(0.15\), it follows that the probability distribution of \(x\) (the number of trustworthy FBI agents tested until someone fails the test ) is: \(P(X = x) = (1-p)^{x-1}p\), where \(P(X = x)\) denotes the probability that the first failure occurs on the \(x^{th}\) test.

The first false positive occurs on the third trial

By substituting \(x = 3\) and \(p = 0.15\) into the formula from Step 1, we find the probability that the first false positive occurs on the third trial: \(P(X = 3) = (1-0.15)^{3-1} * 0.15 = 0.108375\).

Fewer than Four Tested Before First Fail

To compute the probability that fewer than four are tested before the first false positive, compute the cumulative probability of the first three trials because 'fewer than four' means one, two, or three. The cumulative probability is the sum of the probabilities of the first, second, and third trials, found by invoking the formula from Step 1 for \(x = 1, 2, 3\). Thus: \(P(X < 4) = P(X = 1) + P(X = 2) + P(X = 3)\). Substitute to find \(P(X<4) = 0.15 + (1-0.15)*0.15 + (1-0.15)^2*0.15 = 0.35955\).

More than Three Agents Tested before First Fail

To compute the probability that more than three agents are tested before the first false positive, subtract the the cumulative probability of the first three trials from one. Thus, \(P(X > 3) = 1 - P(X < 4) = 1-0.35955 = 0.64045\). This is because the probability of all possible outcomes equals 1, so the probability of the complementary event to 'fewer than four' (namely, 'more than three') can be found by subtracting from one the probability of 'fewer than four'.

Key concepts

Probability Distribution

To begin with, let's talk about what a probability distribution is. In essence, it gives us a way to list all possible outcomes of a random experiment and the probability that each outcome will occur. When we refer to the geometric distribution, as seen in the polygraph testing scenario, we're talking about a type of probability distribution that applies to scenarios with two possible outcomes for each trial - success or failure - and assumes that the probability of success is constant for each trial.

Now, the geometric distribution specifically deals with the number of trials required to get the first success. Here, 'success' could be a bit misleading because it refers to the first false-positive result from the polygraph test, which is an undesirable outcome. But from a statistical standpoint, this is what we're aiming to find out. The formula that we use for the geometric distribution is:
\[P(X = x) = (1-p)^{x-1}p\]
Where \(P(X = x)\) denotes the probability that the first 'success' (false-positive) occurs on exactly the \(x^{th}\) attempt, \(p\) represents the probability of getting a 'success' on any given trial, and \(1-p\) is the probability of 'failure' (or a true negative in our context).

False-Positive Probability

A false-positive in statistics is an error in data reporting in which a test result improperly indicates the presence of a condition, such as a polygraph test signaling deception when none exists. The probability of a false-positive is the probability that the test incorrectly identifies a non-event as an event. In the polygraph example from the exercise, the false-positive probability is given as 0.15. This means there is a 15% chance that the polygraph will indicate a trustworthy FBI agent is deceptive.

Understanding the false-positive probability is vital, especially when the stakes are high. In real-world applications, such as medical testing or security checks, a high rate of false positives can lead to unnecessary stress, additional tests, and even wrongful suspicion. Therefore, it's crucial to measure and minimize the false-positive rate as much as possible to increase the accuracy and reliability of the testing procedure.

Cumulative Probability

Cumulative probability comes into play when we're interested in the probability of an event occurring by a certain point in time, or in our case, within a certain number of trials. It's essentially 'adding up' the probabilities of individual outcomes to find the likelihood of a collection of outcomes. For our polygraph testing example, we would use cumulative probability to calculate the odds of encountering the first false-positive within the first, second, or third test.

The calculation of cumulative probability involves summing the individual probabilities up to the desired number of trials. For example, with a false-positive rate of 0.15, to find out the cumulative probability up to the third person being tested, you add the probability of getting the first false-positive on the first, second, and third attempts, corresponding to \(P(X < 4)\). This kind of information is helpful when planning and preparing for potential outcomes ahead of time. In practice, knowing cumulative probabilities aids in risk assessment and making informed decisions in areas such as project management, financial planning, or even public health strategies.