When is the probability of A given B equal to probability of B given A? Can they ever equal each other?
If not, then is it because the denominator ($P(A)$ vs $P(B)$) is not the same?
I'm asking because in Probability for the Enthusiastic Beginner (A wonderful book by the way), the author says they aren't equal ... In general.

They can be equal, but it would be a coincidence.  To see they are not equal in general, just think about how conceptually $P(A|B)$ and $P(B|A)$ are different kinds of probabilities.  
To use a standard example: the probability that I test positive given that I have a certain disease (a measure of how accurate the test is in detecting the disease) is a different kind of probability from the probability that I have a certain disease, given that I test positive (which largely depends on the base prevalence of the disease: if the base rate is very low, there might be a good chances we are dealing with a false positive, no matter how accurate the test is)
