The expected value of the life insurance policy to the insurance company can be calculated by analyzing the outcomes of the two possible scenarios: the person survives the year and the person dies within the year.

Step 1: Understanding the problem

• Premium paid by the policyholder: $350 (annual premium)
• Policy payout if the policyholder dies: $150,000 (insurance payout)
• Probability that the policyholder survives the year: $0.997735$
• Probability that the policyholder dies: $1 - 0.997735 = 0.002265$

Step 2: Calculating the outcomes for the insurance company

1. If the policyholder survives:

   • The insurance company receives the premium of $350.
   • The company does not pay any benefit.

   Therefore, the net gain for the insurance company if the policyholder survives is:
   $350$ dollars.

2. If the policyholder dies:

   • The insurance company receives the premium of $350.
   • The company pays the benefit of $150,000.

   Therefore, the net loss for the insurance company if the policyholder dies is:
   $350 - 150,000 = -149,650$ dollars.

Step 3: Calculating the expected value

The expected value is the weighted average of these two outcomes, weighted by their probabilities. The formula for expected value $E(X)$ is:

$E(X) = P(\text{survive}) \times \text{gain from surviving} + P(\text{die}) \times \text{loss from dying}$

Substitute the values:

$E(X) = (0.997735) \times 350 + (0.002265) \times (-149,650)$

$E(X) = 349.19975 - 339.796225$

$E(X) = 9.403525$

Step 4: Interpretation

The expected value of the life insurance policy to the insurance company is $9.40. This means that, on average, the insurance company expects to make a profit of $9.40 per policy sold to a 45-year-old female under these conditions.

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Would you like further clarification or additional examples?

Here are some related questions you may find interesting:

1. How does the expected value change if the policyholder is older or younger?
2. What would happen to the expected value if the premium was higher or lower?
3. How would the expected value change if the probability of survival was different (e.g., if the survival rate was 0.995)?
4. How does the insurance company's overall expected value compare when selling multiple policies?
5. What role does the concept of risk play in life insurance premium calculations?

Tip: The expected value is crucial in determining the profitability of insurance policies and helps companies set premiums that balance risk and reward effectively.