Question

The article "FBI Says Fewer than 25 Failed Polygraph Test" (San Luis Obispo Tribune, July 29,2001 ) states that falsepositives in polygraph tests (tests in which an individual fails even though he or she is telling the truth) are relatively common and occur about \(15 \%\) of the time. Let \(x\) be the number of trustworthy FBI agents tested until someone fails the test. a. Is the probability distribution of \(x\) binomial or geometric? b. What is the probability that the first false-positive will occur when the third agent 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 distribution is Geometric. b. The probability that the first false-positive will occur when the third agent is tested is 0.108375. c. The probability that fewer than four are tested before the first false-positive occurs is 0.385875. d. The probability that more than three agents are tested before the first false-positive occurs is 0.614125.

Step-by-step solution

Identifying The Distribution

This case describes a situation where FBI agents are tested repeatedly until a the first false-positive result appears. This is a typical scenario of a geometric distribution, since we are counting the number of trials till we get our first success (false-positive result). Hence, probability distribution of \(x\) is geometric.

Calculating The Probability of First False-Positive at Third Trial

In a geometric distribution, the probability that the first success will occur on the nth trial is given by \(P(X=n) = p * (1-p)^{n-1}\).Here the probability of getting a false-positive on a single trial \(p = 0.15\). Therefore, the probability that first false positive will occur when the third agent is tested is given by \(P(X=3) = 0.15 * (1-0.15)^{3-1} = 0.15 * (0.85)^2 = 0.108375\)

Calculating The Probability of First False-Positive Before Fourth Trial

The probability that the first success occurs before the fourth trial is given by adding up the probabilities of the first success occurring on the first, second and third trials i.e. \(P(X<4) = P(X=1) + P(X=2) + P(X=3)\) = \(0.15*1 + 0.15*0.85 + 0.15*(0.85)^2 = 0.15 + 0.1275 + 0.108375 = 0.385875\)

Calculating The Probability of First False-Positive After Third Trial

To calculate the probability that more than three agents are tested before the first false-positive occurs, we subtract the probability that a success occurs within the first three trials from 1. It's easier to find the probability of the opposite event and subtract that from 1. This is given by \(1 - P(X < 4) = 1 - 0.385875 = 0.614125\)