If $P(A) \neq Q(A)$, then is $P(A|B) \neq Q(A|B)$? [closed]
The title pretty much sums it up. Although I'm curious as to the general answer, I'm specifically referring to two different $P_\theta,\, \theta \in \Theta$ from a family of probability measures $(P = \{P_\theta\})$ on a statistical experiment with space $(\mathcal X,\mathcal B, P)$. So, for any $B \in \mathcal B$, if $P_{\theta_1}\neq P_{\theta_2}$, does that mean $P_{\theta_1}(A|B) \neq P_{\theta_2}(A|B)$?
Solution 1:
Or take $B=A$ to get $P(A\mid B) = Q(A\mid B) = 1$ even if $P(A) \ne Q(A)$.