Suppose a life insurance company sells a $\$ 270,000$ 1-year term life insurance policy to a 20 -year-old female for $\$ 190$. 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 $\$ 66.88$.
(Round to the nearest cent as needed.)
Which of the following interpretations of the expected value is correct? Select the correct choice below and fill in the answer box to complete your choice.
(Round to the nearest cent as needed.)
A. The insur 1 month.
on every 20 -year-old female it insures for
The insurance company expects to make a maximum profit of $\$ \square$ on every 20 -year-old female it insures for B. 1 year.

C. The insurance company expects to make a profit of $\$$ on every 20 -year-old female it insures for 1 month.
D. The insurance company expects to make a profit of $\$ \square$ on every 20 -year-old female it insures for 1 year.

Step 1 ：The expected value of a random variable is the long-term average or mean value. It is the sum of all possible values each multiplied by the probability of its occurrence.

Step 2 ：In this case, the insurance company either makes a profit of $190 (the cost of the policy) if the female survives the year, or it loses $270,000 - $190 = $269,810 if the female does not survive the year.

Step 3 ：The probabilities of these events are given as 0.999544 and 1 - 0.999544 = 0.000456 respectively.

Step 4 ：We can calculate the expected value by multiplying each outcome by its probability and summing these products. The outcomes are $190 and -$269810, and the probabilities are 0.999544 and 0.000456 respectively.

Step 5 ：The expected value of the policy to the insurance company is approximately $66.88. This means that, on average, the insurance company expects to make a profit of $66.88 on every 20-year-old female it insures for 1 year.

Step 6 ：This is the long-term average profit the company expects to make per policy sold to a 20-year-old female, taking into account both the probability of the female surviving the year and the probability of her not surviving the year.

Step 7 ：Final Answer: The correct interpretation of the expected value is that the insurance company expects to make a profit of \(\boxed{66.88}\) on every 20 -year-old female it insures for 1 year.