BBBASICSTAT8 6.2.021

Question

Aldrich Ames is a convicted traitor who leaked American secrets to a foreign power. Yet Ames took routine lie detector tests and each time passed them. How can this be done? Recognizing control questions, employing unusual breathing patterns, biting one's tongue at the right time, pressing one's toes hard to the floor, and counting backwards by 7 are countermeasures that are difficult to detect but can change the results of a polygraph examination†. In fact, it is reported in Professor Ford's book that after only 20 minutes of instruction by "Buzz" Fay (a prison inmate), 85% of those trained were able to pass the polygraph examination even when guilty of a crime. Suppose that a random sample of nine students (in a psychology laboratory) are told a "secret" and then given instructions on how to pass the polygraph examination without revealing their knowledge of the secret. What are the following probabilities? (Round your answers to three decimal places.)

(a) all the students are able to pass the polygraph examination

(b) more than half the students are able to pass the polygraph examination

(c) no more than half of the students are able to pass the polygraph examination

(d) all the students fail the polygraph examination

Answer 1

Assume that each of 9 students can pass the polygraph examination with probability p =0.85 and independently of ecah other. Then number of the students who are able to pass the polygraph examination has binomial distribution and

(a) P(all the students are able to pass the polygraph examination) = p⁹

(b) P(more than half the students are able to pass the polygraph examination) = ∑_k=5⁹ C₉^k p^k(1-p)^9-k

(c) P(no more than half of the students are able to pass the polygraph examination)= ∑_k=0⁵ C₉^k p^k(1-p)^9-k

(d) P(all the students fail the polygraph examination) = (1-p)⁹