To calculate the expected value for the amount of claims, we use the formula for expected value:

$E(X) = \sum \text{(Value of Claim)} \times \text{(Probability of Claim)}$

Using the values and probabilities from the table:

• $0 \times 0.65 = 0$
• $50,000 \times 0.26 = 13,000$
• $100,000 \times 0.05 = 5,000$
• $150,000 \times 0.02 = 3,000$
• $200,000 \times 0.01 = 2,000$
• $250,000 \times 0.01 = 2,500$

Now summing these:

$E(X) = 0 + 13,000 + 5,000 + 3,000 + 2,000 + 2,500 = 25,500$

The expected value of the claims is $25,500.

Would you like a detailed explanation of this process or help with parts b and c? Here are some related questions to explore further:

1. What is the interpretation of the expected value in this context?
2. How does the distribution of probabilities affect the expected value?
3. What is the variance and standard deviation of the claim amounts?
4. If the probabilities change, how would the expected value be recalculated?
5. How would you use this expected value in insurance decision-making?

Tip: When working with probabilities, always ensure their sum equals 1 to validate the data.