Sure! Let's break down this problem step-by-step to understand how the expected return from the insurance company is calculated.
1. Define the Variables:
- The insured amount (coverage) is \[tex]$15,000.
- The annual premium paid is \$[/tex]150.
- The empirical probability of the painting being stolen (theft) is 0.03.
- The probability that the painting will not be stolen is therefore [tex]\(1 - 0.03 = 0.97\)[/tex].
2. Calculate the Payout in Each Scenario:
- If the painting is stolen, the insurance company pays out \[tex]$15,000.
- If the painting is not stolen, no payout is made and you only lose the premium paid.
3. Expected Value Calculation:
The expected return from the insurance can be calculated using the concept of expected value \(E(X)\), which is a weighted average of all possible outcomes, where each outcome is weighted by its probability of occurrence.
4. Calculate Total Expected Payout:
- Expected payout if theft occurs: \(\text{Coverage} \times \text{Probability of theft} = 15,000 \times 0.03\).
- Expected loss from paying the premium when no theft occurs: \(\text{-Premium} \times \text{Probability of no theft} = -150 \times 0.97\).
5. Combine These Values to Find the Expected Return:
\[
E(X) = (\text{Coverage} \times \text{Probability of theft}) - (\text{Premium} \times \text{Probability of no theft})
\]
Plugging in the values we have:
\[
E(X) = (15,000 \times 0.03) - (150 \times 0.97)
\]
6. Calculate the Numerical Values:
- \(15,000 \times 0.03 = 450\)
- \(150 \times 0.97 = 145.5\)
7. Subtract the Expected Loss from the Expected Payout:
\[
E(X) = 450 - 145.5 = 304.5 \text{ dollars}
\]
Therefore, the expected return from the insurance company if you take out this insurance is \$[/tex]304.5.
