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The annual premium for a [tex]\$15,000[/tex] insurance policy against the theft of a painting is [tex]\$150[/tex]. If the empirical probability that the painting will be stolen during the year is [tex]0.03[/tex], what is your expected return from the insurance company if you take out this insurance?

Let [tex]X[/tex] be the random variable for the amount of money received from the insurance company in the given year.

Calculate [tex]E(X)[/tex]:

[tex]
E(X) = \$304.5 \text{ dollars}
[/tex]


Sagot :

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.