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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.

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Two events are mutually exclusive if $P(A\cap B)=0$. This means that there is no way for these two events to occur simultaneously.

Two events are independent if $P(A)\cdot P(B)=P(A\cap B)$, but $P(A\cap B)$ isn't necessarily zero.

In your example let $E$ be the event of an emergency room claim and $O$ be the event of an operating room claim. We are told that $P(E\cup O)=0.9$, $P(E^C)=0.25$, and that events $E$ and $O$ are independent. This tells us only that $$ P(E\cap O)=P(E)\cdot P(O). $$ Since $P(E^C)=0.25$, then we must have $P(E)=1-P(E^C)=1-0.25=0.75$. This can be put into our formula to give us $$ P(E\cap O)=0.75\cdot P(O). $$ Then you can use sum rule for probabilities to finish this question off and get $P(O)=0.6$.

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