Let us suppose that you are undergoing a series of mammography scans in order to rule out possible breast cancer cases. We define the sensitivity for a positive event by the variable X . It takes binary values with X=1 representing a positive event and X=0 being a negative event. We reserve Y as a classification parameter for either a negative or a positive breast cancer confirmation. (Short note on wordings: positive here means having breast cancer, although none of us would consider this being a positive thing).
We let Y=1 represent the the case of having breast cancer and Y=0 as not.
Let us assume that if you have breast cancer, the test will be positive with a probability of 0.8 , that is we have
p(X=1\vert Y=1) =0.8.This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of 80\% for having cancer. It is however not correct, as the following Bayesian analysis shows.