SOLUTION: 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? Recogn

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Question 1206492: 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 five 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 is the probability that no more than half of the students are able to pass the polygraph examination?
Answer by Shin123(626) About Me  (Show Source):
You can put this solution on YOUR website!
No more than half of the students means 0, 1, or 2 students passed. This is a binomial distribution. The probability is
0.15%5E5%2B5%2A0.85%2A0.15%5E4%2B10%2A0.85%5E2%2A0.15%5E3.
The 0.15%5E5 term represents the probability that none of the students passed. Since each student as a 85% pass rate, they have a 15% fail rate.
The 5%2A0.85%2A0.15%5E4 term represents the probability that 1 student passed. The 0.85%2A0.15%5E4 is the probability that 1 student passes and 4 fail, and it is multiplied by 5 because there are 5 possible ways to pick a student that passes.
The final term 10%2A0.85%5E2%2A0.15%5E3 is for 2 students passing. Again, the 0.85%5E2%2A0.15%5E3 term is the probability that any 2 given students pass and the other 3 fail. There are 5C2=10 ways to pick these 2 students.
Plugging this into a calculator, we get the probability as 0.0266 (rounded to 4 decimal places).
If you need more precision, plug it into a calculator yourself either online or on an actual calculator.