Problem 56 - Algebra for College Students

Question

Sandra finds that the annual premium for a \(\$ 2000\) insurance policy against the theft of a painting is \(\$ 100\). If the probability that the painting will be stolen during the year is \(0.01\), what is Sandra's expected gain or loss in taking out the insurance? \(-\$ 80\)

Step-by-Step Solution

Answer

Sandra's expected loss is $80.

Step 1: Determine Potential Outcomes

There are two potential outcomes for Sandra: the painting is either stolen or it is not. If the painting is stolen, Sandra receives $2000 from the insurance. If it is not, she loses her premium of $100.

Step 2: Calculate Expected Loss if Stolen

The probability that the painting is stolen is 0.01. If stolen, Sandra gains $2000 (the insurance policy amount) but forfeits her $100 premium, resulting in a net gain of $2000 - $100 = $1900.

Step 3: Calculate Expected Loss if Not Stolen

The probability that the painting is not stolen is 0.99. If not stolen, Sandra simply loses her premium of $100.

Step 4: Compute Expected Value

The expected value (or expected gain/loss) can be calculated by multiplying each outcome by its probability and summing the results: \[Expected\ Gain/Loss = (0.01 \times 1900) + (0.99 \times -100) \]Which simplifies to:\[Expected\ Gain/Loss = 19 - 99 = -80 \]

Step 5: Interpret the Result

An expected loss of $80 means on average, Sandra would lose $80 per year by insuring the painting, based on the given probabilities and premiums.

Key Concepts

Insurance Premium Calculation

Insurance premium calculation is pivotal for determining the cost of insuring an item, like Sandra's painting. The insurance premium is the amount paid by the policyholder to the insurer for coverage. In this context, Sandra pays an annual premium of $100 for a $2000 insurance policy. This premium needs to be carefully set, taking into account the potential risks involved.
When insurers calculate premiums, they assess various factors:

• Value of the item being insured
• Likelihood of the insured event occurring (in this case, theft)
• Potential losses the insurer might incur

Through these calculations, insurance companies aim to offer premiums that are fair to both parties, making it viable for the insurer to cover possible claims while still being affordable for the insured. In Sandra's case, the premium of $100 is determined based on the risk of theft and the value of the painting.

Probability of Events

Understanding the probability of events is crucial when evaluating scenarios such as Sandra's insurance decision. The probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In probability terms, a value of 0 represents an impossible event, while 1 denotes certainty.
In Sandra's situation, the probability that her painting gets stolen is 0.01, or 1 in 100, indicating a rather low risk. Conversely, the probability that the painting remains safe is 0.99, or 99 out of 100, which is quite high. Each of these probabilities influences her expected gain or loss from the insurance.
For insurance purposes, accurately estimating these probabilities is essential. Misjudging them can lead to incorrect premium settings and financial implications for both the insurer and the insured. Probabilities help in understanding and balancing potential risks and rewards, ensuring informed decision-making.

Net Gain or Loss Calculation

Calculating the net gain or loss from an investment or decision like insurance involves evaluating possible outcomes and their respective probabilities. For Sandra, this means considering the financial implications of her painting either being stolen or remaining safe.
The calculation of expected gain or loss involves:

• The financial outcome if the painting is stolen: a gain of \(1900 (\)2000 insurance payout minus \(100 premium).
• The outcome if the painting isn't stolen: a loss of \)100 (the premium paid).

Sandra's expected gain or loss is calculated using the formula:\[Expected\ Gain/Loss = (Probability\ of\ Theft \times Gain) + (Probability\ of\ No\ Theft \times Loss)\]Substitute the values:\[Expected\ Gain/Loss = (0.01 \times 1900) + (0.99 \times -100)\]This results in an expected loss of \(80. This calculation helps Sandra understand that, on average, she would be losing \)80 annually by taking the insurance, which helps her make an informed decision about the policy's value.