The article "FBI Says Fewer than 25 Failed Polygraph Test" (San Luis Obispo Tribune, July 29, 2001) states that false-positives 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. Suppose that such a test is given to 10 trustworthy individuals.
a. What is the probability that all 10 pass?
b. What is the probability that more than two fail, even though all are trustworthy?
c. The article indicated that 500 FBI agents were required to take a polygraph test. Consider the random variable x = number of the 500 tested who fail. If all 500 agents tested are trustworthy, what are the mean and standard deviation of x?
d. The headline indicates that fewer than 25 of the 500 agents tested failed the test. Is this a surprising result if all 500 are trustworthy? Answer based on the values of the mean and standard deviation from Part (c).