> P(cancer | positive_mammogram) = P(positive_mammogram | cancer)

Sorry if I've got this wrong, but does this line say that we're assuming the probability of someone having cancer if they've got a positive mammogram, is the same as the probability of them having a positive mammogram if they've got cancer?

As far as I know, the first one would be the Positive Predictive Value (PPV) of the mammogram, the second one would be the Sensitivity of the mammogram.

They're related but not usually the same. The PPV would change depending on the prevalence of the disease (go up as the prevalence goes up), but the sensitivity would remain the same.