An insurance company estimates the probability of an earthquake in the next year to be 0.0013.  The average damage done by an earthquake it estimates to be $60,000.  If the company offers earthquake insurance for $100, what is their expected value of the policy?

The expected value (EV) of the policy for the insurance company is calculated by considering two scenarios:

1. **If an earthquake occurs**: The company will lose the payout for damages, minus the premium received.
2. **If no earthquake occurs**: The company will only gain the premium received, without any additional costs.

### Let's break this down:

1. **Probability of an earthquake**: $ P(\text{earthquake}) = 0.0013 $
2. **Damage from an earthquake**: $ \text{Damage} = 60,000 $
3. **Premium charged**: $ \text{Premium} = 100 $

#### Scenario 1: Earthquake Occurs
If an earthquake occurs, the company loses the damage amount minus the premium received:
\[
\text{Loss} = 60,000 - 100 = 59,900
\]
The probability of this event is 0.0013.

#### Scenario 2: No Earthquake
If no earthquake occurs, the company simply gains the premium, which is $100$. The probability of this event is:
\[
P(\text{no earthquake}) = 1 - 0.0013 = 0.9987
\]

### Expected Value (EV) Calculation
The EV is the sum of the outcomes weighted by their probabilities:
\[
\text{EV} = (0.0013 \times (-59,900)) + (0.9987 \times 100)
\]

Let's calculate this.

\[
\text{EV} = (0.0013 \times (-59,900)) + (0.9987 \times 100)
\]
\[
\text{EV} = -77.87 + 99.87
\]
\[
\text{EV} = 22
\]

### Conclusion:
The expected value of the policy for the insurance company is **$22**. This means, on average, the company expects to make a profit of $22 per policy sold.

Would you like further details on this or have any questions?

Here are 5 related questions for further exploration:
1. How would the expected value change if the probability of an earthquake increases?
2. What happens to the expected value if the average damage is higher than estimated?
3. How can the insurance company adjust the premium to ensure a higher profit margin?
4. If the number of policyholders increases, how does the law of large numbers affect the company’s overall expected profit?
5. How would reinsurance affect the expected value calculation for the insurance company?

**Tip:** In risk management, adjusting premiums and understanding probabilities are key to long-term profitability in insurance companies.

An insurance company estimates the probability of an earthquake in the next year to be 0.0013. The average damage done by an earthquake it estimates to be $60,000. If the company offers earthquake insurance for $100, what is their expected value of the policy?

Discover how to calculate the expected value of an earthquake insurance policy using probability and expected value formulas. This problem demonstrates how to assess risk in insurance and is suitable for students in Grades 10-12.

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