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Question 14, 5.1.32
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There is a 0.99965 probability that a randomly selected 22-year-old female lives through the year. An insurance company wants to offer her a one-year policy with a death benefit of $900,000. How much should the company charge for this policy if it wants an expected return of $300 from all similar policies?
The company should charge $ (Round to the nearest dollar.)

To solve this problem, we need to calculate how much the insurance company should charge for the policy, given that they want to have an expected return of $300 from all similar policies. Here’s how we approach the problem step by step:

### Definitions:
- Probability of living through the year: $ P(\text{live}) = 0.99965 $
- Probability of death during the year: $ P(\text{death}) = 1 - P(\text{live}) = 1 - 0.99965 = 0.00035 $
- Death benefit: $900,000
- Expected return per policy: $300

Let the premium that the company charges be denoted by $ x $ dollars.

### Step 1: Calculate Expected Payout
The expected payout of the policy is the amount the company expects to pay, weighted by the probability of death. Since the company only pays out if the person dies, the expected payout is:

\[
\text{Expected Payout} = P(\text{death}) \times \text{Death Benefit} = 0.00035 \times 900,000 = 315
\]

### Step 2: Set Up Expected Profit Equation
The company wants an expected return of $300. Therefore, the company needs to charge a premium that compensates for the expected payout plus the desired return. The expected return equation is:

\[
\text{Expected Premium} - \text{Expected Payout} = 300
\]

\[
x - 315 = 300
\]

### Step 3: Solve for $ x $

\[
x = 300 + 315 = 615
\]

### Final Answer:
The company should charge **$615** for the policy.

Would you like further details or clarifications? Here are 5 related questions to explore this further:

1. What if the death benefit was increased to $1,000,000, how would that affect the premium?
2. How does changing the expected return (e.g., from $300 to $500) affect the premium charged?
3. How would the premium change if the probability of surviving decreased slightly?
4. What is the significance of expected return in the context of insurance policies?
5. How would the company's profit change if more people survived than expected?

**Tip:** Understanding expected value helps in many real-life scenarios, especially in decision-making under uncertainty, like insurance!

There is a 0.99965 probability that a randomly selected 22-year-old female lives through the year. An insurance company wants to offer her a one-year policy with a death benefit of $900,000. How much should the company charge for this policy if it wants an expected return of $300 from all similar policies? The company should charge $ (Round to the nearest dollar.)

Learn how to calculate the premium for a one-year life insurance policy, given a 0.99965 survival probability and a $900,000 death benefit, with the insurance company aiming for an expected return of $300. This problem covers probability, expected value, and algebraic calculations, suitable for advanced high school or college-level students.

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