What is the probability that a randomly selected woman will have a positive mammogram?

8.19 A sensor is used to monitor the performance of a nuclear reactor. The sensor accurately reflects the state of the reactor with a probability of .97. But with a probability of .02, it gives a false alarm (by reporting excessive radiation even though the reactor is performing normally), and with a probability of .01, it misses excessive radiation (by failing to report excessive radiation even though the reactor is performing abnormally).

 

(a) What is the probability that a sensor will give an incorrect report, that is, either a false alarm or a miss?

 

(b) To reduce costly shutdowns caused by false alarms, management intro-duces a second completely independent sensor, and the reactor is shut down only when both sensors report excessive radiation. (According to this perspective, solitary reports of excessive radiation should be viewed as false alarms and ignored, since both sensors provide accurate information much of the time.) What is the new probability that the reactor will be shut down because of simultaneous false alarms by both the first and second sensors?

 

(c) Being more concerned about failures to detect excessive radiation, someone who lives near the nuclear reactor proposes an entirely different strategy: Shut down the reactor whenever either sensor reports excessive radiation. (According to this point of view, even a solitary report of excessive radiation should trigger a shutdown, since a failure to detect excessive radiation is potentially cata-strophic.) If this policy were adopted, what is the new probability that excessive radiation will be missed simultaneously by both the first and second sensors?

 

8.21 Assume that the probability of breast cancer equals .01 for women in the 50–59 age group. Furthermore, if a woman does have breast cancer, the probability of a true positive mammogram (correct detection of breast cancer) equals .80 and the probability of a false negative mammogram (a miss) equals .20. On the other hand, if a woman does not have breast cancer, the probability of a true negative mammogram (correct nondetection) equals .90 and the probability of a false positive mammogram (a false alarm) equals .10. (Hint: Use a frequency analysis to answer questions. To facilitate checking your answers with those in the book, begin with a total of 1,000 women, then branch into the number of women who do or do not have breast cancer, and finally, under each of these numbers, branch into the number of women with positive and negative mammograms.)

 

(a) What is the probability that a randomly selected woman will have a positive mammogram?

 

(b) What is the probability of having breast cancer, given a positive mammogram?

 

(c) What is the probability of not having breast cancer, given a negative mammogram?