Using conditional probability, or bayes theorem to calculate a conditional probabilitiy, does not make something bayesian analysis.
I haven't read the original report (it's on my list) but what is presented in this article isn't really anything except probability. It would be bayesian if they put a distribution on the proportion of people that have a "certain cancer."
John Allen Paulos seems to be very intent on increasing the numeric literacy (I think he calls it numeracy) of people in general. Its a laudable goal indeed.
This is the kind of example that we can give to grade school kids and have them work it out. It would be a great "ah ha!" moment for some of them.
