The table shows claims and their probabilities for an insurance company.
Amount of claim 
​(to the nearest​ $20,000)
Probability
​$0
0.70
​$20,000
0.18
​$40,000
0.08
​$60,000
0.02
​$80,000
0.01
​$100,000
0.01
 
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Part 1
a.  Calculate the expected value.
​$
  
enter your response here

To calculate the expected value, we use the formula for expected value $ E(X) $ of a discrete random variable:

\[
E(X) = \sum \left( \text{Amount of Claim} \times \text{Probability} \right)
\]

Given the dataTo calculate the expected value, we use the formula for expected value $ E(X) $ of a discrete random variable:

\[
E(X) = \sum \left( \text{Amount of Claim} \times \text{Probability} \right)
\]

Given the 

- $ \$0 $ with probability $ 0.70 $
- $ \$20,000 $ with probability $ 0.18 $
- $ \$40,000 $ with probability $ 0.08 $
- $ \$60,000 $ with probability $ 0.02 $
- $ \$80,000 $ with probability $ 0.01 $
- $ \$100,000 $ with probability $ 0.01 $

Let's calculate the expected value:

\[
E(X) = (0 \times 0.70) + (20,000 \times 0.18) + (40,000 \times 0.08) + (60,000 \times 0.02) + (80,000 \times 0.01) + (100,000 \times 0.01)
\]

I'll compute this now.# Defining the amounts and their corresponding probabilities
amounts = [0, 20000, 40000, 60000, 80000, 100000]
probabilities = [0.70, 0.18, 0.08, 0.02, 0.01, 0.01]

# Calculating the expected value
expected_value = sum(amount * prob for amount, prob in zip(amounts, probabilities))
expected_valueThe expected value of the claims is \$9,800.

Would you like more details or have any questions? 

Here are 5 related questions for further practice:
1. How does changing the probability of a \$40,000 claim affect the expected value?
2. What is the variance of this distribution?
3. What would happen to the expected value if the probability of a \$0 claim decreased?
4. How would you compute the standard deviation for this set of claims?
5. If the probability of a \$100,000 claim increased, how would that affect the risk to the insurance company?

**Tip:** The expected value is a measure of the center or "average" of a probability distribution. It's useful in assessing long-term outcomes in insurance and other probabilistic situations.

The table shows claims and their probabilities for an insurance company. Amount of claim (to the nearest $20,000) Probability $0 0.70 $20,000 0.18 $40,000 0.08 $60,000 0.02 $80,000 0.01 $100,000 0.01 Part 1 a. Calculate the expected value.

Learn how to calculate the expected value of insurance claims based on probabilities. This problem covers key concepts in probability and expected value, ideal for students in Grades 10-12. Understand the impact of different claim amounts and their associated probabilities on the overall expected value.

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