Breast cancer is the most common form of cancer in women, affecting about 10% of women at some time in their lives. There is about a 1% chance of having breast cancer at a given time (that is, P(S) = 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 P(S and POS) = 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 P(POS) = 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 P(S | POS) 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 explain why this suggests that P(S | POS) is only about 0.08.

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Breast Diagnostic Cancer Test Pos (1) Yes (1) 100 Women_ Pos (12) No (99) Neg (87