Question

according to a life insurance companys mortality table, the probability that a woman of a certain age will survive 1 year is 0.992 and the probability that she will die within 1 year is 0.008. if she buys a \\$50,000, 1-year policy for \\$500, what is the companys expected gain or loss?

select the correct choice below and, if necessary, fill in the answer box within your choice.

a. the company has an expected loss of \\$ \\_
b. the company has an expected gain of \\$ \\_
c. the company has neither an expected gain nor an expected loss.

Explanation:

Define the random variable and outcomes

Using the Expected Value and Actuarial Science knowledge points
Let \(X\) represent the company's net financial gain.

• Outcome 1 (Woman survives):
• Probability: \(P(\text{survive}) = 0.992\)
• Gain: \(x_1 = \$500\) (premium kept)
• Outcome 2 (Woman dies):
• Probability: \(P(\text{die}) = 0.008\)
• Gain: \(x_2 = \$500 - \$50,000 = -\$49,500\) (premium minus payout)

Calculate the expected value

Using the Expected Value knowledge point

$$
LATEXBLOCK0
$$

Interpret the result

Using the Expected Value knowledge point
Since \(E(X) = 100 > 0\), the company has an expected gain of \(\$100\).

Answer:

• A. The company has an expected loss of $ <blank></blank>.
• B. The company has an expected gain of $ <blank>100</blank>. (Correct answer)
• C. The company has neither an expected gain nor an expected loss.