5.57 Mammogram diagnostics Breast cancer is the most common form of cancer in women, affecting about 10%

Question:

of women at some time in their lives. There is about a 1%

chance of having breast cancer at a given time (that is, P1S2 = 0.01 for the state of having breast cancer at a given time). The chance of breast cancer increases as a woman ages, and the American Cancer Society recommends an annual mammogram after age 40 to test for its presence. Of the women who undergo mammograms at any given time, about 1% are typically estimated to actually have breast cancer.

The likelihood of a false test result varies according to the breast density and the radiologist’s level of experience. For use of the mammogram to detect breast cancer, typical values reported are sensitivity = 0.86 and specificity = 0.88.

a. Construct a tree diagram in which the first set of branches shows whether a woman has breast cancer and the second set of branches shows the mammogram result. At the end of the final set of branches, show that P1S and POS2 = 0.01 * 0.86 = 0.0086 and report the other intersection probabilities also.

b. Restricting your attention to the two paths that have a positive test result, show that P1POS2 = 0.1274.

c. Of the women who receive a positive mammogram result, what proportion actually have breast cancer?

d. The following tree diagram illustrates how P1S •POS2 can be so small, using a typical group of 100 women who have a mammogram. Explain how to get the frequencies shown on the branches and why this suggests that P1S •POS2 is only about 0.08.

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