Suppose a life insurance company sells a $\$ 240,000$ 1-year term life insurance policy to a 20 -year-old female for $\$ 230$. According to the National Vital Statistics Report, $58(21)$, the probability that the female survives the year is 0.999544. Compute and interpret the expected value of this policy to the insurance company.
The expected value is $\$ \square$.
(Round to the nearest cent as needed.)
Answer
Final Answer: The expected value of this policy to the insurance company is \(\boxed{\$120.56}\). This means that, on average, the insurance company expects to make a profit of \$120.56 from each policy sold.
Step 1 ：Let's denote the profit that the insurance company makes from selling the policy as a random variable. There are two possible outcomes:
Step 2 ：1. The female survives the year. The probability of this happening is 0.999544. In this case, the insurance company makes a profit of \$230 (the cost of the policy).
Step 3 ：2. The female does not survive the year. The probability of this happening is 1 - 0.999544 = 0.000456. In this case, the insurance company makes a loss of \$240,000 (the payout of the policy) - \$230 (the cost of the policy) = -\$239,770.
Step 4 ：The expected value is then the sum of the profits in each case, weighted by the probability of each case happening. So, the expected value is (0.999544 * \$230) + (0.000456 * -\$239,770) = \$120.56
Step 5 ：Final Answer: The expected value of this policy to the insurance company is \(\boxed{\$120.56}\). This means that, on average, the insurance company expects to make a profit of \$120.56 from each policy sold.
