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:
$\Pr(E \cap O) = \Pr(E) * \Pr(O)\tag{1}$
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):
$\Pr(E \cup O) = \Pr(E) + \Pr(O) – \Pr(E \cap O)\tag{2}$
substitute (1) into (2).
$\Pr(O) = \cfrac{\Pr(E \cup O) – \Pr(E)}{\Pr(E')}= .4$
I appreciate any help. Thank you!
Best Answer
The "or hospital" is unfortunate wording, and should be disregarded.
For calculations, there are many ways. I would probably note that $\Pr(O' \cap E')=0.15$. Then from $\Pr(O')\Pr(E')=0.15$ we find that $\Pr(O')=0.6$, and hence $\Pr(O)=0.4$. Your calculation may be a little more efficient.