The annual premium for a

​$5 comma 0005,000

insurance policy against the theft of a painting is

​$300300.

If the​ (empirical) probability that the painting will be stolen during the year is

0. 03.03​,

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 find the expected return from the insurance company, we need to calculate the expected value $ E(X) $ of the random variable $ X $, which represents the amount of money received from the insurance company. 

Here, $ X $ can take on two values:
1. $ X = 0 $ if the painting is not stolen.
2. $ X = 5000 $ if the painting is stolen.

The probability $ P $ that the painting will be stolen during the year is given as 0.03, and the probability that it will not be stolen is $ 1 - 0.03 = 0.97 $.

The expected value $ E(X) $ is calculated as follows:

\[ E(X) = (0 \times 0.97) + (5000 \times 0.03) \]

\[ E(X) = 0 + 150 \]

\[ E(X) = 150 \text{ dollars} \]

Thus, the expected return from the insurance company is $\$150$.

If you have any questions or want more details, let me know. Here are 5 related questions you might ask:

1. How do you calculate expected value in probability?
2. What is the significance of expected value in insurance?
3. Can you explain other examples of expected value in real-life scenarios?
4. How does probability affect insurance premiums?
5. What is the difference between empirical and theoretical probability?

**Tip:** Always verify the given probabilities and conditions to ensure accurate calculations in probability-related problems.

Learn how to calculate the expected return from an insurance policy using probability concepts. This problem involves determining the expected value of receiving insurance payout based on theft probability. Suitable for advanced high school students and anyone interested in probability in insurance.

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