4-3 Complements, Conditional Probability, and Bayes’ Theorem 179 b. Find the probability of randomly selecting a student who kept the money, given that the student was given a $1 bill. c. What do the preceding results suggest? 15. Denomination Effect a. Find the probability of randomly selecting a student who spent the money, given that the student was given four quarters. b. Find the probability of randomly selecting a student who spent the money, given that the student was given a $1 bill. c. What do the preceding results suggest? 16. Denomination Effect a. Find the probability of randomly selecting a student who kept the money, given that the student was given four quarters. b. Find the probability of randomly selecting a student who kept the money, given that the student was given a $1 bill. c. What do the preceding results suggest? In Exercises 17–20, refer to the accompanying table showing results from experiments conducted by researchers Charles R. Honts (Boise State University) and Gordon H. Barland (Department of Defense Polygraph Institute). In each case, it was known whether or not the subject lied, so the table indicates when the polygraph (lie detector) test was correct. Subject Did Not Lie Subject Lied Polygraph indicated that the subject lied. 15 42 Polygraph indicated that the subject did not lie. 32 9 17. False Positive Find the probability of selecting a subject with a positive polygraph test result, given that the subject did not lie. Why is this particular case problematic for test subjects? 18. False Negative Find the probability of selecting a subject with a negative polygraph result, given that the subject lied. What would be an unfavorable consequence of this error? 19. Positive Predictive Value Find the positive predictive value for the test. That is, find the probability that a subject lied, given that the polygraph indicated lying. Does the result make the polygraph appear to be effective? 20. Negative Predictive Value Find the negative predictive value for the polygraph. That is, find the probability that a subject did not lie, given that the polygraph indicates that the subject did not lie. Does the result make the test appear to be effective? 21. Redundancy in Computer Hard Drives The Seagate ST8000NM0055 hard drive has a 1.22% rate of failures in a year (based on data from Backblaze, Inc.). For the following, assume that all hard drives are that Seagate model. a. If all of your computer data are stored on a hard disk drive with a copy stored on a second hard disk drive, what is the probability that during a year, you can avoid catastrophe with at least one working drive? Express the result with six decimal places. b. If copies of all of your computer data are stored on three independent hard disk drives, what is the probability that during a year, you can avoid catastrophe with at least one working drive? Express the result with six decimal places. What is wrong with using the usual round-off rule for probabilities in this case?
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