I am renewing my probability knowledge and I am having trouble trying to solve some exercises.
- 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 on hospital claims.
Calculate the probability that a claim submitted to the insurance company includes operating room charges.
My thinking here is that the percentage of claims including operating room charges could range from $10$% up to $85$%. $10$% happens when there are no claims including emergency room and operating room charges together, $85$% happens when $75$% of the total claims include emergency room and operating room charges together and $10$% of the total claims include only operating room charges.
From here I would deduce that the probability is $\frac{10+85}{2}=47.5$% but the answer is wrong. It should be $40$%. Can anyone help?
Best Answer
Let $E$ be the event of a claim including emergency room, and $O$ be the event of a claim including operating room. From the second information, $$\Pr(E') = 25\%, \ \Pr(E) = 75\%$$ Then using the first information, and the independence of $E$ and $O$, $$\begin{align*} \Pr(E\cup O) =& 85\%\\ \Pr(E)+\Pr(O) - \Pr(E\cap O) =& 85\%\\ \Pr(E) + \Pr(O) - \Pr(E)\Pr(O) = & 85\%\\ 75\% + \Pr(O) - 75\%\cdot\Pr(O) =& 85\%\\ \Pr(O) =& 40\% \end{align*}$$
Using less mathematical approach, draw a table of two events: $$\begin{array}{c|c|c|c} &E&E'&\text{sum}\\ \hline O&a&b&x\\\hline O'&c&d&1-x\\\hline\hline \text{sum}&75\%&25\%&100\% \end{array}$$ From this, all we know more is that $a+b+c = 85\%$, which gives $b=10\%$ considering $a+c=75\%$.
From independence between $O$ and $E$, $$\begin{align*} x:100\% =& b:25\%\\ x =& 2:5\\ =&40\%\end{align*}$$