Math Problem Statement

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

  1. 03.03​,

what is your expected return from the insurance company if you take out this​ insurance?

Question content area bottom

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

Solution

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

Here, XX can take on two values:

  1. X=0X = 0 if the painting is not stolen.
  2. X=5000X = 5000 if the painting is stolen.

The probability PP that the painting will be stolen during the year is given as 0.03, and the probability that it will not be stolen is 10.03=0.971 - 0.03 = 0.97.

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

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

E(X)=0+150E(X) = 0 + 150

E(X)=150 dollarsE(X) = 150 \text{ dollars}

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

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Math Problem Analysis

Mathematical Concepts

Probability
Expected Value

Formulas

Expected value formula E(X) = ∑ (x * P(X=x))

Theorems

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Suitable Grade Level

Advanced High School