A 31-year-old woman purchases a $300,000 term life insurance policy for an annual payment of $570 . Based on a period life table for the U.S. government, the probability that she will survive the year is 0.999055. Find the expected value of the policy for the insurance company. Round to two decimal places for currency problems. The expected value of the policy for the insurance company is?

The expected value of the life insurance policy for the insurance company can be computed using the following formula:

\[
\text{Expected Value} = P(\text{survive}) \times (\text{premium}) + P(\text{not survive}) \times (\text{loss for insurance company})
\]

Where:
- $ P(\text{survive}) $ is the probability the woman survives the year.
- $ P(\text{not survive}) = 1 - P(\text{survive}) $ is the probability the woman does not survive the year.
- $\text{premium}$ is the amount the woman pays annually ($570).
- $\text{loss for insurance company}$ is the amount the insurance company would have to pay out if she does not survive, which is the policy amount ($300,000).

### Step 1: Identify the given values.
- $ P(\text{survive}) = 0.999055 $
- Premium = $570
- Policy payout = $300,000

### Step 2: Calculate $ P(\text{not survive}) $.
\[
P(\text{not survive}) = 1 - 0.999055 = 0.000945
\]

### Step 3: Apply the formula for expected value.

\[
\text{Expected Value} = 0.999055 \times 570 + 0.000945 \times (-300,000)
\]

Let's compute this value.

\[
\text{Expected Value} = (0.999055 \times 570) + (0.000945 \times -300,000)
\]

Now we can calculate the final result.The expected value of the policy for the insurance company is approximately **$285.96**.

Would you like further details or have any questions?

Here are five related questions you might explore:
1. How would the expected value change if the annual premium were different?
2. What if the probability of survival changed due to the woman's age or health?
3. How would increasing the policy amount affect the expected value?
4. Can expected value be used to compare multiple insurance policies?
5. How do insurance companies use expected value in their business models?

**Tip:** When calculating expected values for large data sets, understanding the risk (probability of loss) is crucial in financial planning.

A 31-year-old woman purchases a $300,000 term life insurance policy for an annual payment of $570. Based on a period life table for the U.S. government, the probability that she will survive the year is 0.999055. Find the expected value of the policy for the insurance company. Round to two decimal places for currency problems. The expected value of the policy for the insurance company is?

Calculate the expected value of a $300,000 term life insurance policy for a 31-year-old woman with a 0.999055 probability of survival. Understand key concepts such as probability and expected value in the context of insurance, using relevant formulas and the law of total expectation. Ideal for college-level students in statistics or finance.

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