If A and B are events such that P(A mid B)=P(B mid A), then :

$P(A | B) = P(B | A)$

$=\frac{P(A \cap B)}{P(B)}=\frac{P(A \cap B)}{P(A)}$, $P(A) \neq 0$, $P(B) \neq 0$

as  $P(X | Y) = \frac{X \cap Y}{P(Y)}$

so  $\frac{1}{P(B)}=\frac{1}{P(A)}$

$\Rightarrow P(A) = P(B)$