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 10th year. The probability that only the wife will survive at least 10 years is 0.025, the probability that only the husband will survive at least 10 years is 0.01, and the probability that both of them will survive at least 10 years is 0.96. what is the expected excess of premiums over claims, given that the husband survives at least 10 years?
I understood that it was a conditional probability problem but I tried to solve it a different way and I'm not sure where my thinking is wrong. Here's what I tried:
X: wife survives 10 years, Y: husband survives 10 years
Since we are assuming the husband survives the 10 years, I disregarded the probability that only the wife survives the 10 years.
-(Premium for both policies) + (Death Benefit For One) * (Probability Only Husband Survives) + (No Death Benefit) * (Probability Husband and Wife Both Survive) $$-1000 + 10000\times P(X' \cap Y) +0\times P(X \cap Y)$$
I have the solution so I know how to "correctly" solve it, I'm just confused about why this method I tried won't work. Thanks in advance!
You want the conditional expectation: $-1000+10000\,\mathsf P(X'\mid Y)+0\,\mathsf P(X\mid Y)$.
You are given $\mathsf P(X'\cap Y)=0.01$ and $\mathsf P(X\cap Y)=0.96$.
Recall the definition of conditional probability: $$\begin{align}\mathsf P(X\mid Y)&=\dfrac{\mathsf P(X\cap Y)}{\mathsf P(Y)}\\[2ex]\mathsf P(X'\mid Y)&=\dfrac{\mathsf P(X'\cap Y)}{\mathsf P(Y)}\end{align}$$
Also recall the law of total probability : $$\mathsf P(Y)=\mathsf P(X\cap Y)+\mathsf P(X'\cap Y)$$