Expected Value for Life Insurance There is a 0.9968 probability that a randomly selected 50-year-old female lives through the year (based on data from the U.S. Department of Health and Human Services). A Fidelity life insurance company charges \(226 for insuring that the female will live through the year. If she does not survive the year, the policy pays out \)50,000 as a death benefit.

a. From the perspective of the 50-year-old female, what are the values corresponding to the two events of surviving the year and not surviving.

b. If a 50-year-old female purchases the policy, what is her expected value?

c. Can the insurance company expect to make a profit from many such policies? Why

The probability of a 50-year oldfemale living through the year is equal to 0.9968.

The insurance company charges $226 for one year and gives $50,000 on the death of a woman.

a.

The monetary value of the event of surviving through the year is equal to -$226 as the charges of the insurance policy need to be paid.

Corresponding to the event of not surviving, any particular woman will receive $50,000.

Therefore, the corresponding monetary value is equal to the following:

$50,000-226=49,774\text{dollars}$

Thus, the monetary value of the event not surviving is equal to $49,774.

b.

The expected value of purchasing the insurance policy is equal to the following:

$$\begin{gathered} \text{Expected}\text{value}=\left(\text{Net}\text{amount} \\ \text{received}\right)\times \left(\text{Probability}\text{of} \\ \text{not}\text{surviving}\right)-\left(\text{Net}\text{amount} \\ \text{given}\right)\times \left(\text{Probability} \\ \text{of}\text{surviving}\right) \\ =\left(49774\right)\left(1-0.9968\right)-\left(226\right)\left(0.9968\right) \\ =-66\text{dollars} \end{gathered}$$

Thus, the expected value of purchasing an insurance policy is equal to -66 dollars.

c.

Since on each policy sold, the insurance company earns 66 dollars, the insurance company makes a lot of profit by selling many such policies.