Chapter 15

A private college report contains these statistics: \(70 \%\) of incoming freshmen attended public schools. \(75 \%\) of public school students who enroll as freshmen eventually graduate. \(90 \%\) of other freshmen eventually graduate. a) Is there any evidence that a freshman's chances to graduate may depend upon what kind of high school the student attended? Explain. b) What percent of freshmen eventually graduate?

A company's records indicate that on any given day about \(1 \%\) of their day- shift employees and \(2 \%\) of the night-shift employees will miss work. Sixty percent of the employees work the day shift. a) Is absenteeism independent of shift worked? Explain. b) What percent of employees are absent on any given day?

Suppose that \(23 \%\) of adults smoke cigarettes. It's known that \(57 \%\) of smokers and \(13 \%\) of nonsmokers develop a certain lung condition by age 60 . a) Explain how these statistics indicate that lung condition and smoking are not independent. b) What's the probability that a randomly selected 60 -year-old has this lung condition?

Police often set up sobriety checkpointsroadblocks where drivers are asked a few brief questions to allow the officer to judge whether or not the person may have been drinking. If the officer does not suspect a problem, drivers are released to go on their way. Otherwise, drivers are detained for a Breathalyzer test that will determine whether or not they will be arrested. The police say that based on the brief initial stop, trained officers can make the right decision \(80 \%\) of the time. Suppose the police operate a sobriety checkpoint after 9 p.m. on a Saturday night, a time when national traffic safety experts suspect that about \(12 \%\) of drivers have been drinking. a) You are stopped at the checkpoint and, of course, have not been drinking. What's the probability that you are detained for further testing? b) What's the probability that any given driver will be detained? c) What's the probability that a driver who is detained has actually been drinking? d) What's the probability that a driver who was released had actually been drinking?

An airline offers discounted "advancepurchase" fares to customers who buy tickets more than 30 days before travel and charges "regular" fares for tickets purchased during those last 30 days. The company has noticed that \(60 \%\) of its customers take advantage of the advance-purchase fares. The "no- show" rate among people who paid regular fares is \(30 \%\), but only \(5 \%\) of customers with advance-purchase tickets are no-shows. a) What percent of all ticket holders are no-shows? b) What's the probability that a customer who didn't show had an advance- purchase ticket? c) Is being a no-show independent of the type of ticket a passenger holds? Explain.

Dan's Diner employs three dishwashers. Al washes \(40 \%\) of the dishes and breaks only \(1 \%\) of those he handles. Betty and Chuck each wash \(30 \%\) of the dishes, and Betty breaks only \(1 \%\) of hers, but Chuck breaks \(3 \%\) of the dishes he washes. (He, of course, will need a new job soon. ....) You go to Dan's for supper one night and hear a dish break at the sink. What's the probability that Chuck is on the job?

A company manufacturing electronic components for home entertainment systems buys electrical connectors from three suppliers. The company prefers to use supplier A because only \(1 \%\) of those connectors prove to be defective, but supplier A can deliver only \(70 \%\) of the connectors needed. The company must also purchase connectors from two other suppliers, \(20 \%\) from supplien \(B\) and the rest from supplier \(\mathrm{C}\). The rates of defective connectors from \(\mathrm{B}\) and \(\mathrm{C}\) are \(2 \%\) and \(4 \%\), respectively. You buy one of these components, and when you try to use it you find that the connector is defective. What's the probability that your component came from supplier A?

In July 2005 the journal Annals of Internal Medicine published a report on the reliability of HIV testing. Results of a large study suggested that among people with HIV, \(99.7 \%\) of tests conducted were (correctly) positive, while for people without HIV \(98.5 \%\) of the tests were (correctly) negative. A clinic serving an at-risk population offers free HIV testing, believing that \(15 \%\) of the patients may actually carry HIV. What's the probability that a patient testing negative is truly free of HIV?

Lie detectors are controversial instruments, barred from use as evidence in many courts. Nonetheless, many employers use lie detector screening as part of their hiring process in the hope that they can avoid hiring people who might be dishonest. There has been some research, but no agreement, about the reliability of polygraph tests. Based on this research, suppose that a polygraph can detect \(65 \%\) of lies, but incorrectly identifies \(15 \%\) of true statements as lies. A certain company believes that \(95 \%\) of its job applicants are trustworthy. The company gives everyone a polygraph test, asking, "Have you ever stolen anything from your place of work?" Naturally, all the applicants answer \({ }^{\prime} \mathrm{No}\)," but the polygraph identifies some of those answers as lies, making the person ineligible for a job. What's the probability that a job applicant rejected under suspicion of dishonesty was actually trustworthy?