An insurance company pays hospital claims. The number of claims that include emergency room or operating room charges is 85% 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 or hospital claims. Calculate the probability that a claim to the insurance company includes operating room charges.
From the question stem, I interpreted the given data as follows: Let E = Emergency room charges, and O= operating room charges $\Pr(E \cup O) = 0.85$ $\Pr(E') = 0.25$ $\Pr(E) = 0.75$ The part of the question stem that confused me is this statement:
The occurrence of emergency room charges is independent of the occurrence of operating room charges or hospital claims.
Why is it necessary to say or hospital claims? At first, I understood that there were two sets, namely Emergency room charges and Operating room charges, but when they state or hospital claims, it makes it seem as if there is a third set when in fact it is just the main space encompassing everything. So you really should get: $\Pr(E \cap O) = \Pr(E) * \Pr(O)\tag$ When they say or hospital claims, it makes me want to write: $\Pr(E \cap O \cap H) = \Pr(E) * \Pr(O) * \Pr(H)$. Is there something about independence I don't understand? Is it really necessary to state this in the problem? Once you got (1), I understand you simply plug it in the inclusion exclusion equation and solve for Pr(O): $\Pr(E \cup O) = \Pr(E) + \Pr(O) - \Pr(E \cap O)\tag$ substitute (1) into (2). $\Pr(O) = \cfrac<\Pr(E \cup O) - \Pr(E)><\Pr(E')>= .4$ I appreciate any help. Thank you!