Allan, oh **, the elementary math in my previous comment is completely wrong. (In the scenario I gave, the probability that you have breast cancer is 1%, not 10%, before taking the test.) My argument doesn’t even approximately work as given: if having breast cancer makes it more likely that you get a positive mammography, then indeed getting a positive mammography must make it more likely that you have breast cancer. Sorry!
(I’m still convinced that my argument re the LHC is correct, but I realize that I’m just looking stupid right now, so I’ll just shut up for now :-))
Allan, oh **, the elementary math in my previous comment is completely wrong. (In the scenario I gave, the probability that you have breast cancer is 1%, not 10%, before taking the test.) My argument doesn’t even approximately work as given: if having breast cancer makes it more likely that you get a positive mammography, then indeed getting a positive mammography must make it more likely that you have breast cancer. Sorry!
(I’m still convinced that my argument re the LHC is correct, but I realize that I’m just looking stupid right now, so I’ll just shut up for now :-))