Complement of conditional probability (trivial probability question)

Question

This is a trivial question but I just wanted to check whether:

If $P(A|B) = 0.90$ then $P(A^C|B) = 0.10$

Answer 1

Indeed, we have the complement rule for conditional probability.

The prove proceeds as follow:

$$P(A^c|B)=\frac{P(A^c\cap B)}{P(B)}=\frac{P(B)-P(A\cap B)}{P(B)}=1-\frac{P(A\cap B)}{P(B)}=1-P(A|B).$$