Math Problem Statement
The probability distribution for damage claims paid by the Newton Automobile Insurance Company on collision insurance follows.
Payment ($) Probability 0 0.85 500 0.04 1,000 0.04 3,000 0.03 5,000 0.02 8,000 0.01 10,000 0.01
(a) Use the expected collision payment to determine the collision insurance premium that would enable the company to break even. If required, round your answers to two decimal places. If your answer is zero, enter "0". x ($) f(x) xf(x) ($) 0 500 1000 3000 5000 8000 10000 Total
Solution
To find the collision insurance premium that would allow the Newton Automobile Insurance Company to break even, we need to calculate the expected collision payment, . The expected value is calculated using the formula:
where represents the payment amounts and represents the probabilities.
Let's calculate it step by step:
Given table:
\hline \text{Payment (x)} & \text{Probability (f(x))} \\ \hline 0 & 0.85 \\ 500 & 0.04 \\ 1,000 & 0.04 \\ 3,000 & 0.03 \\ 5,000 & 0.02 \\ 8,000 & 0.01 \\ 10,000 & 0.01 \\ \hline \end{array}$$ ### Now, calculate $$x \cdot f(x)$$: $$\begin{array}{|c|c|c|} \hline \text{Payment (x)} & \text{Probability (f(x))} & x \cdot f(x) \\ \hline 0 & 0.85 & 0 \times 0.85 = 0 \\ 500 & 0.04 & 500 \times 0.04 = 20 \\ 1,000 & 0.04 & 1,000 \times 0.04 = 40 \\ 3,000 & 0.03 & 3,000 \times 0.03 = 90 \\ 5,000 & 0.02 & 5,000 \times 0.02 = 100 \\ 8,000 & 0.01 & 8,000 \times 0.01 = 80 \\ 10,000 & 0.01 & 10,000 \times 0.01 = 100 \\ \hline \end{array}$$ ### Sum of all $$x \cdot f(x)$$: $$0 + 20 + 40 + 90 + 100 + 80 + 100 = 430$$ Thus, the expected collision payment, $$E(X)$$, is **$430**. ### Conclusion: To break even, the insurance company should set the collision insurance premium to at least **$430**. Do you want any more details or have any questions? Here are 5 related questions that might expand your understanding: 1. What is the significance of the expected value in insurance policies? 2. How would changes in probabilities affect the expected collision payment? 3. What other costs might the insurance company need to consider besides the break-even premium? 4. How could the company use variance to assess the risk of collision payments? 5. What happens if the actual claims differ significantly from the expected value? **Tip:** When working with probabilities, ensure that the sum of the probabilities equals 1 to avoid calculation errors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Distribution
Expected Value
Insurance Premium
Formulas
E(X) = Σ(x_i * f(x_i))
Theorems
Expected Value Theorem
Suitable Grade Level
Grades 11-12
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