Based on current estimates of how well mammograms detect breast cancer, Table 8.19 shows what to expect for 100,000 adult women over the age of 40 in terms of whether a woman has breast cancer and whether a mammogram gives a positive result (i.e., indicates that the woman has breast cancer).

(a) Construct the conditional distributions for the mammogram test result, given the true disease status. Does the mammogram appear to be a good diagnostic tool?

(b) Construct the conditional distribution of disease status, for those who have a positive test result. Use this to explain why even a good diagnostic test can have a high false positive rate when a disease is not common.