Question
A medical test has been designed to detect the presence of a certain disease. Among people who have the disease, the probability that the disease will be detected by the test is $.95 .$ However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is $.04$. It is estimated that $4 \%$ of the population who take this test have the disease. a. If the test administered to an individual is positive, what is the probability that the
A medical test has been designed to detect the presence of a certain disease. Among people who have the disease, the probability that the disease will be detected by the test is $.95 .$ However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is $.04$. It is estimated that $4 \%$ of the population who take this test have the disease. a. If the test administered to an individual is positive, what is the probability that the person actually has the disease? b. If an individual takes the test twice and the test is positive both times, what is the probability that the person actually has the disease? (Assume that the tests are independent.)
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Components arriving at a distributor are checked for defects by two different inspectors (each component is checked by both inspectors. The first inspector detects 90$\%$ of all defectives that are present, and the second inspector does likewise. At least one inspector fails to detect a defect on 20$\%$ of all defective components. What is the probability that the following occur?
(a) A defective component will be detected only by the first inspector? By exactly one of the two inspectors?
(b) All three defective components in a batch escape detection by both inspectors (assuming inspections of different components are independent of one another)?
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