Taking 50/50 as a prior indicates that you don't know the true prevalence of C.
Note that here 0.9 is not a probability, but rather a degree of belief. In this concrete example it tells you how much you should be worried upon observing R. If the prevalence of C is unknown (even if C is actually rare!), then given a positive R one should be worried a lot, which is exactly what 0.9 says.
Also note the language change and its implications for policy making, etc: Instead of saying "there is [not enough] evidence that .." we say "given the observations and these assumptions we should [not] believe/expect that .."
All of this is true. I was writing to mainly illustrate that Bayesian analysis is still open to abuse—though as the sibling commenter says, it's hopefully clearer when it's being abused.
Note that here 0.9 is not a probability, but rather a degree of belief. In this concrete example it tells you how much you should be worried upon observing R. If the prevalence of C is unknown (even if C is actually rare!), then given a positive R one should be worried a lot, which is exactly what 0.9 says.
Also note the language change and its implications for policy making, etc: Instead of saying "there is [not enough] evidence that .." we say "given the observations and these assumptions we should [not] believe/expect that .."