One thing I don't understand for Bayes' theorem is that I end up in a circular position sometimes when I try to solve it. $P(B|C) = \frac{P(C\mid B) \times P(B)}{P(C)}$ However, I don't know $P(C\mid B)$!
If I try to use Bayes' theorem to calculate it, then I end up right back where I started.
I know $P(B)$ and $P(C)$, but they are dependent on a variable $A$ whose probability I also know. How do I get out of the circle if I don't have a prior for $P(C\mid B)$?