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.