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.
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$\begingroup$Suppose $P(A),P(B),P(A\cap B)\neq 0$. Then $$ P(A|B)=P(B|A) \Leftrightarrow \frac{P(A\cap B)}{P(B)}=\frac{P(A\cap B)}{P(A)} \Leftrightarrow P(A)=P(B). $$
$\endgroup$ 3 $\begingroup$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)
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