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The problem is asking for the expected value of an insurance policy for the insurance company. Here's how to break it down:

### Given Data:
- The premium the company charges for the policy: $1300.
- The probability of a claim for $5000 is 0.09.
- The probability of a total loss of $150,000 is 0.004.
- If neither of these claims happen, the company earns the full premium.

### Approach:
To calculate the expected value to the insurance company, we consider all possible outcomes and their probabilities:

1. **No Claim**: 
   - Probability = $1 - (0.09 + 0.004) = 0.906$
   - The company gains the full premium: $1300.

2. **Claim for $5000**: 
   - Probability = 0.09.
   - The company receives the premium ($1300) but pays out $5000.
   - Net result for the company: $1300 - 5000 = -3700$.

3. **Total Loss of $150,000**: 
   - Probability = 0.004.
   - The company receives the premium ($1300) but pays out $150,000.
   - Net result for the company: $1300 - 150000 = -148700$.

### Expected Value Calculation:
The expected value (EV) is calculated by summing up the products of each outcome and its corresponding probability.

\[
EV = (0.906 \times 1300) + (0.09 \times -3700) + (0.004 \times -148700)
\]

Now, let's calculate it.The expected value of this insurance policy to the company is approximately $250.00.

Would you like more details or have any questions?

Here are 5 follow-up questions:
1. How does increasing the premium affect the expected value?
2. What would the expected value be if the probability of a claim for $5000 increased to 0.1?
3. How can insurance companies use expected value to set their premiums?
4. What is the significance of expected value in risk management?
5. How does changing the payout amount for claims affect the overall expected value?

**Tip:** In risk-related decisions, understanding the expected value helps balance potential losses against regular gains like premiums.

ACME insurance company estimates that the probability of the owners' of a house worth between $120,000 to $170,000 making a claim for $5000 is 0.09, and that the probability of the house and contents being totally destroyed is 0.0040. Should that tragedy happen, the company will have to pay $150,000. The company charges $1300 for the insurance policy. What is the expected value of this policy to the insurance company? Round your answer to two decimal places.

Calculate the expected value of an insurance policy using probabilities of claims for $5000 and total loss of $150,000. This solution involves applying the expected value formula, using probabilities of 0.09 and 0.004 for claims, and considering the $1300 premium. Ideal for students learning probability and insurance mathematics.

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