Consider two events, $A$ and $B$, with $P(A) \neq 0$ and $P(B) \neq 0$.
If $P(A|B) = P(B|A)$, why is $P(A) \neq P(B)$?
I thought that $P(A|B) = \frac {P(A\cap B)} {P(B)}$ and $P(B|A) = \frac {P(B\cap A)} {P(A)}$?
Then $P(A\cap B) = P(B\cap A)$, so wouldn't $P(A) = P(B)$?
The problem is you cannot cancel $P(A\cap B)$ when it is $0$. $A$ and $B$ could be disjoint with positive but different probabilities. Of course $P(A|B) =P(B|A)$ does not imply that $P(A) \neq P(B)$. Nor does it imply that $P(A)=P(B)$.