The annual premium for a ​$10 comma 000 insurance policy against the theft of a painting is ​$200. If the​ (empirical) probability that the painting will be stolen during the year is 0. 02​, what is your expected return from the insurance company if you take out this​ insurance?
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Part 1
Let X be the random variable for the amount of money received from the insurance company in the given year. 
​E(X)equals
  
enter your response here dollars

To determine your expected return from the insurance company, we need to calculate the expected value of the random variable $ X $, which represents the amount of money you receive from the insurance company in the given year.

Here’s how we can compute it:

1. **Amount to be received if the painting is stolen**: If the painting is stolen, you would receive the insurance payout of $10,000.
   
2. **Amount to be received if the painting is not stolen**: If the painting is not stolen, you would receive $0 from the insurance company.

3. **Empirical probability of the painting being stolen**: $ P(\text{stolen}) = 0.02 $

4. **Empirical probability of the painting not being stolen**: $ P(\text{not stolen}) = 1 - 0.02 = 0.98 $

Now, we can calculate the expected value $ E(X) $ using the formula for expected value of a discrete random variable:

\[
E(X) = ( \text{Payout if stolen} \times P(\text{stolen})) + ( \text{Payout if not stolen} \times P(\text{not stolen}) )
\]

Substituting the values:

\[
E(X) = (10,000 \times 0.02) + (0 \times 0.98)
\]

\[
E(X) = 200 + 0 = 200 \text{ dollars}
\]

Thus, your expected return from the insurance company if you take out this insurance is **200 dollars**.

### Follow-up questions:
1. How would your expected return change if the empirical probability of theft increased to 0.05?
2. What would be the expected return if the insurance payout was only $5,000 for theft?
3. How does the premium amount affect the expected return from an insurance policy?
4. Can you calculate the expected profit or loss for the insurance company based on these figures?
5. What factors might influence the empirical probability of theft for valuable paintings?

**Tip:** The expected return assumes the law of large numbers—over many instances, the actual returns will converge toward the expected value.

The annual premium for a ​$10,000 insurance policy against the theft of a painting is ​$200. If the (empirical) probability that the painting will be stolen during the year is 0.02, what is your expected return from the insurance company if you take out this insurance?

Calculate the expected return from a $10,000 insurance policy with a 2% probability of theft and a $200 premium. This problem involves key concepts such as probability, expected value, and the expected value theorem, ideal for high school students in Grades 10-12.

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