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  	


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, $ E(X) $. The expected value is calculated using the formula:

\[
E(X) = \sum (x_i \times f(x_i))
\]
where $ x_i $ represents the payment amounts and $ f(x_i) $ represents the probabilities.

Let's calculate it step by step:

### Given table:

\[
\begin{array}{|c|c|}
\hline
\text{Payment (x)} & \text{Probability (f(x))} \\
\hline
0 & 0.85 \\
500 & 0.04 \\
1,000 & 0.04 \\
3,000 & 0.03 \\
5,000 & 0.02 \\
8,000 & 0.01 \\
10,000 & 0.01 \\
\hline
\end{array}
\]

### 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.

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'.

Learn how to calculate the expected value of damage claims for the Newton Automobile Insurance Company and determine the break-even collision insurance premium. This example covers probability distribution and expected value calculations, suitable for students in Grades 11-12.

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