The annual premium for a $\$ 20,000$ insurance policy against the theft of a painting is $\$ 200$. If the (empirical) probability that the painting will be stolen during the year is 0.03 , what is your expected return from the insurance company if you take out this insurance?
Let $\mathrm{X}$ be the random variable for the amount of money received from the insurance company in the given year.
\[
E(X)=\square \text { dollars }
\]

Step 1 ：Let \(X\) be the random variable for the amount of money received from the insurance company in the given year. The expected return from the insurance company can be calculated by multiplying the payout of the insurance policy by the probability of the event (theft of the painting) occurring, and then subtracting the cost of the insurance premium.

Step 2 ：In this case, the payout of the insurance policy is $20,000, the probability of the event occurring is 0.03, and the cost of the insurance premium is $200.

Step 3 ：So, the expected return from the insurance company is: \(E(X) = (Payout \times Probability) - Premium\)

Step 4 ：Substituting the given values, we get \(E(X) = (20000 \times 0.03) - 200 = 400\)

Step 5 ：The expected return from the insurance company is $400. This means that, on average, you would expect to receive $400 from the insurance company over the course of a year, after accounting for the cost of the insurance premium.

Step 6 ：Final Answer: \(E(X) = \boxed{400}\)