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This problem involves expected value calculations for an insurance company based on probabilities of a customer making a claim.

### Information given:
- The insurance company charges $1600 annually.
- There's a **10% chance** that the customer will make a claim on a policy of **$5000** (meaning the company loses $3400).
- There's a **90% chance** that the customer won't make a claim (the company keeps the full $1600).

### Expected Value Calculation:
1. **If a claim is made** (10% chance):
   - The insurance company earns $1600 but pays out $5000, resulting in a **loss of $3400**.
   - The probability of this happening is 10% or 0.10.

2. **If no claim is made** (90% chance):
   - The company keeps the $1600.
   - The probability of this happening is 90% or 0.90.

### Formula:
The expected value (E) can be calculated as:
\[
E = (P_{\text{claim}} \times \text{Net Loss with claim}) + (P_{\text{no claim}} \times \text{Profit with no claim})
\]
Where:
- $ P_{\text{claim}} = 0.10 $
- $ P_{\text{no claim}} = 0.90 $
- **Net Loss with claim** = -$3400 (loss)
- **Profit with no claim** = $1600 (profit)

Substituting the values:
\[
E = (0.10 \times -3400) + (0.90 \times 1600)
\]
Let's calculate this.The expected value for the insurance company on this policy is **$1100**. 

Therefore, the correct answer is **(c) $1100**.

Would you like further details on any aspect, or do you have more questions?

### Related Questions:
1. How does the expected value change if the claim probability increases to 15%?
2. What would the expected value be if the insurance policy cost $2000 instead of $1600?
3. How can insurance companies use expected value to determine pricing strategies?
4. What is the role of probability in calculating risk for an insurance company?
5. How would the expected value be different if the payout was $6000 instead of $5000?

### Tip:
When calculating expected values, always consider both possible outcomes and weigh them by their probabilities. This helps assess the overall risk or reward.

An insurance company charges a customer $1600 per year for a particular customer's auto insurance. The company has predicted that there is a 10% chance the person will make a claim on the policy of $5000 (which means the insurance company would lose $3400) and a 90% chance that they won't make a claim. What can the insurance company on average expect to make on selling this policy?

Learn how to calculate the expected value for an insurance policy using probability concepts. This problem involves understanding the chance of a claim and its impact on profit. Suitable for high school students studying financial mathematics and probability.

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