Conditional probabilities and the addition rule

Question

Suppose I want to calculate the probability $P(\text{A or B})$, then I can make use of the addition formula:

$$P(\text{A or B}) = P(A) + P(B) - P(\text{A and B})$$

We subtract $P(\text{A and B})$ in order to avoid double counting.

Now my question is how to apply this formula in the case of a conditional probability is known.

Assume we know that $P(A) = x$ and $P(B) = y$ and there is a conditional probability known as $P(B|A)=z$.

For calculating $P(\text{A or B})$, do I need to take the conditional probability somehow into account, or can I just calculate:

$$P(\text{A or B}) = P(A) + P(B) - P(\text{A and B}) = x + y - (x \cdot y)$$

Answer 1

We have: $$\mathsf P(A\cap B)=\mathsf P(B\mid A)\mathsf P(A)$$ by definition of conditional probability, so that:

$$\mathsf P(A\cup B)=\mathsf P(A)+\mathsf P(B)-\mathsf P(A\cap B)=\mathsf P(A)+\mathsf P(B)-\mathsf P(B\mid A)\mathsf P(A)=x+y-zx$$

Answer 2

$P(A\cap B)=P(A)\cdot P(B)$ if and only if $A$ and $B$ are independent.

By Bayes' Theorem

$$\mathsf P(B\mid A)=\frac{\mathsf P(A\cap B)}{\mathsf P(A)}$$

$$\Rightarrow \mathsf P(A\cap B)=\mathsf P(B\mid A)\cdot \mathsf P(A)$$