It's my understanding the multiplication rule for dependent events is the following:
$$P(A \cap B) = P(A)P(B\mid A) = P(B)P(A\mid B)$$
For independent events:
$$P(B\mid A) = P(B)$$
$$P(A \cap B) = P(A)P(B)$$
I'm getting confused from the set theoretic definition because some authors use this rule with separate sample spaces for event $A$ and $B$. It's my understanding that the two events must intersect $A \cap B$. With two separate disjoint sample spaces the intersection would be $P(\emptyset)$. Is this practice correct? For example:
Here is another example of the sample space being changed. I'm unsure how $P(B|A) = P(B \cap A)/P(A)$ would be calculated because of change in sample space. Also, why is this problem calculated as if the events were independent vs using the just mentioned formula?
