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Let me interpret the probability distribution table and find the correct values for standard deviation. Here's the given information:

### Probability Distribution Table
| Outcome               | Profit $ X = x $ (in $) | Probability $ P(X = x) $ |
|-----------------------|--------------------------|----------------------------|
| House destroyed        | $-20810$               | 0.003                      |
| House survived         | $ 1900 $               | 0.997                      |

### 1. Given:

- The expected value $ E[X] = \mu_X = 1270 $ is correct.
- We need to check the standard deviation calculation $ SD[X] = \sigma_X $, as the previous answer $ 11507 $ was marked incorrect.

### 2. Standard Deviation Calculation:
Standard deviation is calculated using the formula:

\[
\sigma_X = \sqrt{E[X^2] - (E[X])^2}
\]

First, let's compute $ E[X^2] $:

\[
E[X^2] = (-20810)^2 \times 0.003 + (1900)^2 \times 0.997
\]

Then, use this to calculate the variance:

\[
\text{Variance} = E[X^2] - (E[X])^2
\]

Finally, take the square root to get the standard deviation. Let me calculate this for you.The correct standard deviation of the insurance company's profit is approximately $ 1812.6 $ dollars, not the previously calculated $ 11507 $ dollars.

Would you like further explanation on this, or do you have any other questions?

Here are 5 related questions you could explore:
1. How is the expected value useful in real-world insurance calculations?
2. What role does standard deviation play in risk management?
3. How would the results change if the probability of house destruction increased?
4. What happens to the expected profit if the insurance premium changes?
5. How does changing the house's value affect both the expected value and standard deviation?

**Tip:** In risk-related problems, a higher standard deviation indicates more variability, implying a greater level of uncertainty in outcomes.

Nia has purchased the insurance policy to cover the value of her house in case it is destroyed due to fire. The house is worth $210,000, and the premium is $1900. The probability of the house being destroyed is 0.3%, and the insurance company's profit depends on this outcome. Calculate the expected value and standard deviation of the insurance company's profit.

This problem involves calculating the expected value and standard deviation of an insurance company's profit using probability distribution. Nia's house is insured for $210,000 with a 0.3% chance of being destroyed. The insurance premium is $1900, and the company's profit is determined by these factors. Learn how to calculate key financial metrics for insurance scenarios.

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