Here's my question. If I'm given two events, both independent of each other. How do I know if $A \cap B$ is empty or if I have to multiply $Pr(A)\;Pr(B)$ to find $A \cap B$? I had a question as such,
An insurance company pays hospital claims. The number of claims that include emergency room or operating room charges is 90% of the total number of claims. The number of claims that do not include emergency room charges is 25% of the total number of claims. The occurrence of emergency room charges is independent of the occurrence of operating room charges on hospital claims. Calculate the probability that a claim submitted to the insurance company includes operating room charges.
My thinking was $A$ and $B$ have no intersection. But they in fact do. Someone please explain mutually exclusive and independent events to me.