TITLE: Is this always true: $P(A|B) = 1-P(A^c|B)$?
QUESTION [10 upvotes]: Does this identity hold for all events?
$$
P(A|B) = 1-P(A'|B)
$$
Logically speaking, if the probability of $A$ given $B$ occurred is $X$, shouldn't the probability that $A$ does not occur, $A'$, given $B$, be similarly $1-X$?
There is a related question here.
This is the closest that I could get to proving (or disproving) it:

$P(A\cap B)=P(A)-P(A' \cap B)$
$P(A|B)P(B)=P(A)-P(A'|B)P(B)$
$\therefore P(A'|B)= \frac {P(A)} {P(B)}-P(A|B)$

Are there are certain formulae which can be used to prove this? Or does the identity only hold under certain situations, and if so, what kind of situations?
Thanks.

REPLY [5 votes]: The above answer is correct, just for clarification your step 1. in the proof in the question is incorrect.

$P(A\cap B) = P(A) - P(A\cap B')$

or

$P(A\cap B) = P(B) - P(A'\cap B)$

would be true.