Physicians often use two diagnostic tests to determine whether a woman has breast cancer or not. These are the mammographys test and the ultrasound test. (See Globe and Mail for story.) The results of the two tests are independent of one another. For example, if a woman has breast cancer a positive result on one test doesn't increase or decrease the chance of a positive result on the other test. If a woman doesn't have breast cancer then the result on one test doesn't affect the chance of a positive or negative result on the other test. The inciidence rate of breast cancer for women at age 40 or older is 1%. If a woman in this age group has breast cancer, the probability is 80% that she will have a positive mammography result. If she doesn't have breast cancer there is still a 10% chance that she will get a positive mammogram. If a woman 40 or older has breast cancer the chance is 95% that she will have a positive ultrasound test result. On the other hand, if she doesn't have breast cancer then the chance of a positive test result is 4%.
Your patient is a 46 year old woman. She has no family history of breast cancer and she reports that there are no lumps in her breasts. So she can be assumed to have the same chance of having breast cancer -- at least before any information is provided by testing -- as a randomly selected woman over 40 years old. The test results for both the mammography and ultrasound tests have come back. The mammogram and the ulrasound results are both negative.

(1) What hypotheses concerning breast cancer are there about your patient? You should have two.
(2) What are the prior or pretest -- meaning prior to both the mammogram and ultrasound -- or preinvestigation probabilities for each of these hypotheses?
(3) What are the prediction probabilities for a result from the mammogram for each hypothesis and each possible result from the mammography test? You should have four of these.
(4) What is the test result or observational data for your patient? These are not the same for all students in the course. Make sure you get yours right.
(5) What are the posterior or post-mammogram probabilities for each hypothesis? You should have two of these. Show your calculations.
(6) What are the prior or pretest -- now meaning prior to the ultrasound test but posterior to the mammogram test -- or preinvestigation probabilities for each of the hypotheses for the ultrasound test?
(7) What are the prediction probabilities for a result from the ultrasound test for each hypothesis and each possible result from this test? You should have four of these.

(8) What is the test result or observational data for your patient? These are not the same for all students in the course. Make sure you get yours right.
(9) What are the posterior or post-ultrasound probabilities for each hypothesis? You should have two of these. Show your calculations.
(10) The post-ultrasound probabilities for each hypothesis are really the results from both tests combined. State what you will tell your patient about the possibility that she has breast cancer based on the two tests.