To determine the expected value to the insurance company per policy sold and the overall expected profit or loss if 20,000 policies are sold, we’ll break down the calculations as follows.

Step 1: Expected Claim Cost per Policy

Given:

• Each policy sells for $1,200.
• Probabilities of claims:
  • $10,000 claim: 1 in 50 policyholders, or probability $P(10,000) = \frac{1}{50} = 0.02$.
  • $40,000 claim: 1 in 250 policyholders, or probability $P(40,000) = \frac{1}{250} = 0.004$.
  • $60,000 claim: 1 in 500 policyholders, or probability $P(60,000) = \frac{1}{500} = 0.002$.

Step 2: Calculate the Expected Payout per Policy

The expected payout per policy is calculated by multiplying each claim amount by its probability and summing them up.

$\text{Expected Payout} = (10,000 \times 0.02) + (40,000 \times 0.004) + (60,000 \times 0.002)$

Calculating each term:

• $10,000 \times 0.02 = 200$
• $40,000 \times 0.004 = 160$
• $60,000 \times 0.002 = 120$

So, $\text{Expected Payout} = 200 + 160 + 120 = 480$

Step 3: Expected Value per Policy to the Company

The expected value per policy to the company is the difference between the policy sale price and the expected payout.

$\text{Expected Value per Policy} = 1,200 - 480 = 720$

So, the expected value per policy sold is $720.

Step 4: Expected Profit or Loss for 20,000 Policies

If the company sells 20,000 policies, the total expected profit is:

$\text{Expected Profit} = 20,000 \times 720 = 14,400,000$

Final Answers

• The expected value per policy sold is $720.
• The expected profit for 20,000 policies is $14,400,000.

Would you like more details or have any questions?

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1. What is the role of expected value in decision-making for insurance?
2. How would an increase in the cost of a policy affect expected profit?
3. What if the claim probabilities were different? How would that affect profit?
4. Why is the expected value a useful metric in insurance?
5. How do insurance companies use these calculations to assess risk?

Tip: Expected value provides a powerful way to estimate outcomes in scenarios with uncertainty, especially for determining potential profit or loss in business