Questions: Type your answer in the space provided. You may show all of your work in the space provided, or do your work on paper and upload it after the exam. For this problem, you will calculate the expected value for a renter's insurance policy. Here are the facts: - A family of four pays 229 per year for the renters insurance policy (covers the cost of damage and lost items due to theft, fire, etc.). - The average amount of money that the insurance company has to pay when someone files this type of insurance claim is 7000. - The probability that they will not need to use this policy during the year is 0.9754.

Solution Steps

To calculate the expected value of the renter's insurance policy, we need to consider both the cost of the policy and the potential payout. The expected value is calculated by multiplying the probability of each outcome by its corresponding value and summing these products. In this case, we have two outcomes: the family does not file a claim, and the family files a claim. The expected value will help us understand the average financial outcome for the family over time.

Let:

• \( P = 229 \) (annual premium paid by the family)
• \( C = 7000 \) (average claim amount)
• \( P_{\text{no claim}} = 0.9754 \) (probability of no claim)
• \( P_{\text{claim}} = 1 - P_{\text{no claim}} = 0.0246 \) (probability of filing a claim)

The expected value \( E \) can be calculated using the formula: \[ E = (P_{\text{no claim}} \cdot -P) + (P_{\text{claim}} \cdot (C - P)) \] Substituting the values: \[ E = (0.9754 \cdot -229) + (0.0246 \cdot (7000 - 229)) \]

Calculating each term:

1. \( 0.9754 \cdot -229 \approx -223.0006 \)
2. \( 0.0246 \cdot (7000 - 229) = 0.0246 \cdot 6771 \approx 166.2006 \)

Now, summing these results: \[ E \approx -223.0006 + 166.2006 \approx -56.8000 \]

Final Answer