A 31 year old woman purchases a $200,000 term life insurance policy for an annual payment of $360. Based on a life table for the U.S. government, the probability that she will survive the year is 0.999056. Find the expected value of the policy for the insurance company 

To find the expected value of the term life insurance policy for the insurance company, we need to calculate the company's expected gain or loss, which is influenced by both the premium paid and the potential payout (if the policyholder passes away).

### Definitions:
- **Premium paid by the policyholder:** $360 (gain for the company)
- **Death benefit (payout if the policyholder dies):** $200,000 (loss for the company)
- **Probability the policyholder survives the year (no payout):** 0.999056
- **Probability the policyholder dies in the year (payout happens):** $1 - 0.999056 = 0.000944$

### Calculation of Expected Value (EV):
The expected value (EV) for the insurance company is the weighted average of the outcomes based on their probabilities.

1. **If the woman survives:** The company earns the premium and pays nothing.
   - Gain: $360
   - Probability: 0.999056

   Contribution to EV = $ 360 \times 0.999056 = 359.66016 $

2. **If the woman dies:** The company loses the payout minus the premium collected.
   - Loss: $ 200,000 - 360 = 199,640 $
   - Probability: 0.000944

   Contribution to EV = $ -199,640 \times 0.000944 = -188.441216 $

### Total Expected Value:
\[
EV = 359.66016 - 188.441216 = 171.218944
\]

### Final Answer:
The expected value of the policy for the insurance company is approximately **$171.22**.

This means the insurance company expects to earn $171.22 on average for each such policy issued.

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Do you want more details or have any questions about this solution?

Here are five related questions to expand your understanding:
1. How would the expected value change if the probability of survival were slightly lower?
2. What happens to the expected value if the premium amount increases?
3. How does the life expectancy factor affect insurance pricing and expected value?
4. What is the role of risk management in setting life insurance premiums?
5. How can life insurance companies reduce the potential loss from high payouts?

**Tip:** In probability and insurance, the expected value is a critical concept used to balance premiums and risk!

A 31 year old woman purchases a $200,000 term life insurance policy for an annual payment of $360. Based on a life table for the U.S. government, the probability that she will survive the year is 0.999056. Find the expected value of the policy for the insurance company.

Learn how to calculate the expected value of a $200,000 term life insurance policy with an annual premium of $360. Using probability and expected value concepts, this detailed solution explains the potential gains and losses for the insurance company based on a survival probability of 0.999056. Suitable for advanced high school or college-level students.

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