91Ó°ÊÓ

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 supplier \(\mathrm{B}\) 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 \(\mathrm{A}\) ?

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?