To calculate the expected return from the insurance company if you take out the insurance, let's define the random variable [tex]\(X\)[/tex] for the amount of money you'll receive from the insurance company during the year.
Here's a step-by-step guide to solving this problem:
1. Define the Variables:
- Annual Premium (P): The annual cost of the insurance policy is [tex]\( \$150 \)[/tex].
- Policy Coverage Amount (C): The value of the insurance policy coverage is [tex]\( \$10,000 \)[/tex].
- Probability of Theft (T): The probability that your painting will be stolen within the year is [tex]\( 0.01 \)[/tex].
2. Possible Outcomes for [tex]\(X\)[/tex]:
There are two possible outcomes:
- If the painting is stolen, you will receive the policy coverage amount [tex]\(C\)[/tex].
- If the painting is not stolen, you will not receive any payout from the insurance company.
Thus:
- If the painting is stolen, [tex]\( X = \$10,000 \)[/tex].
- If the painting is not stolen, [tex]\( X = \$0 \)[/tex].
3. Corresponding Probabilities:
- Probability that the painting is stolen (T): [tex]\( P(X = 10000) = 0.01 \)[/tex].
- Probability that the painting is not stolen: [tex]\( P(X = 0) = 1 - 0.01 = 0.99 \)[/tex].
4. Expected Value Calculation:
The expected value [tex]\(E(X)\)[/tex] is the sum of all possible values of [tex]\(X\)[/tex] multiplied by their respective probabilities.
[tex]\[
E(X) = (10000 \times 0.01) + (0 \times 0.99)
\][/tex]
Simplify the equation:
[tex]\[
E(X) = 100 + 0 = 100
\][/tex]
Therefore, the expected amount received from the insurance company is [tex]\( \$100 \)[/tex].
5. Calculate the Expected Return:
The expected return is the difference between the expected amount received and the premium paid.
[tex]\[
\text{Expected Return} = \text{Expected Amount Received} - \text{Annual Premium}
\][/tex]
Substitute the values:
[tex]\[
\text{Expected Return} = 100 - 150 = -50
\][/tex]
Thus, your expected return from the insurance company if you take out this insurance policy is [tex]\( \$ -50 \)[/tex].
