To calculate the expected value ($EV$) of the life insurance policy for the insurance company, we consider two possible outcomes: the policyholder survives the year or does not survive the year.

Definitions and Variables:

1. Policy income (premium collected): $350
2. Payout (if the policyholder dies): $160,000
3. Probability of survival ($P_{\text{survive}}$): 0.999544
4. Probability of death ($P_{\text{death}}$): $1 - P_{\text{survive}} = 1 - 0.999544 = 0.000456$

Expected Value Formula:

$EV = (\text{Income if survives} \times P_{\text{survive}}) + (\text{Income if dies} \times P_{\text{death}})$

Step 1: Determine income in each case

• If survives: The company earns $350 (premium), and there is no payout. Thus, income = $350.
• If dies: The company loses $160,000 (payout) minus $350 (premium), so income = $-159,650$.

Step 2: Substitute into the formula

$EV = (350 \times 0.999544) + (-159,650 \times 0.000456)$

Step 3: Compute the components

• $350 \times 0.999544 = 349.8404$
• $-159,650 \times 0.000456 = -72.5994$

$EV = 349.8404 - 72.5994 = 277.241$

Final Answer:

The expected value of this policy to the insurance company is approximately: $\boxed{277.24}$

Interpretation:

The expected value of $277.24 means that, on average, the insurance company expects to earn $277.24 per policy sold to 20-year-old females under these conditions.

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Would you like additional details or clarification on this solution?

Expanding Questions:

1. What is the impact of changing the probability of survival on the expected value?
2. How does a higher premium affect the expected value for the insurance company?
3. What happens to the expected value if the payout amount is reduced?
4. How can this analysis be applied to a larger group of policyholders?
5. What assumptions are made in this calculation, and how do they affect the result?

Tip:

Expected value provides insight into average outcomes but does not guarantee specific results for any individual case.