So here is the typical question: I know that Bayes' Theorem should be applied here, but I don't understand why. I am going to pretend I don't know the formula and try to solve this, but I can't seem to get it: W = Women in age group 40 B = Has Breast cancer P = Positive tested for breast cancer .01W = B (.8B)W = P (.096[W - B])W = P Since 1% of women in this age group have breast cancer, 80% of those with breast cancer in this age group have positive mammographies, and 9.6% of those without breast cancer in the age group also have positive mammographies. I solved for W, B, and P and got 9.333 (repeating), .09333 (repeating), and .0696888 (repeating) respectively. So the probability of having breast cancer from a positive mammography would be: .09333 / .0696888, no? But obviously that isn't the correct answer since that's over 100% chance. How does one work this out intuitively? I don't even understand how the Bayesian formula is derived for this type of problem. I understand that I have to use it and why it works, but not why I have to use it and why I have to apply it.