Two life insurance policies, each with a death benefit of $10,000$ and a one-time premium of $500$, are sold to a couple, one for each person. The policies will expire at the end of the tenth year. The probability that only the wife will survive at least ten years is $0.025$, the probability that only the husband will survive at least ten years is $0.01$, and the probability that both of them will survive at least ten years is $0.96$.
What is the expected excess of premiums over claims, given that the husband survives at least ten years?
The answer my professor provided is: $896.9072165...$
I am having trouble deriving this answer.
My method was as follows:
$\frac{0*(0.96)+10,000*(0.01)+10,000*(0.025)}{0.01+0.96}=\frac{350}{0.97}=360.82847...+500=860.8247423...$
Does this make sense?
No, you have confused the first part.
P(wife survives | husband has survived) $=\dfrac{0.96}{0.96+0.01} = \dfrac{96}{97}$
Expected excess of premiums over claims $= \left(\dfrac{96}{97}\cdot 500 - \dfrac1{97}\cdot9500\right)$ + 500