Question

Insurance. An insurance policy costs $\mathrm{\$}100$ and will pay policyholders $\mathrm{\$}10,000$ if they suffer a major injury (resulting in hospitalization) or $\mathrm{\$}3000$ if they suffer a minor injury (resulting in lost time from work). The company estimates that each year 1 in every 2000 policyholders may have a major injury, and 1 in 500 a minor injury only. a) Create a probability model for the profit on a policy. b) What's the company's expected profit on this policy? c) What's the standard deviation?

Short answer

a) Define outcomes and probabilities. b) Expected profit: $89. c) Standard deviation: $248$.

Step-by-step solution

Define Outcomes and Probabilities

We start by defining possible outcomes and their probabilities. The outcomes include no injury, a minor injury, or a major injury. - **No Injury**: Probability is the complement of the sum of probabilities of injuries.- **Minor Injury**: Probability is given as 1 in 500, which is $\frac{1}{500}=0.002$.- **Major Injury**: Probability is 1 in 2000, which is $\frac{1}{2000}=0.0005$.The probability of no injury is $1-(0.002+0.0005)=0.9975$.

Calculate Profit for Each Outcome

Define the profit from each outcome considering the policy cost and payout:- **No Injury**: The company retains the entire premium, so the profit is $100$.- **Minor Injury**: The company pays $3000$, so the profit is $100-3000=-2900$.- **Major Injury**: The company pays $10,000$, so the profit is $100-10,000=-9900$.

Calculate Expected Profit

To find the expected profit, multiply each outcome's profit by its probability and sum them up:$$E(X)=100(0.9975)+(-2900)(0.002)+(-9900)(0.0005)$$Calculating gives:$$E(X)=99.75-5.8-4.95=89$$The expected profit per policy is $\mathrm{\$}89$.

Calculate Variance and Standard Deviation

First, calculate the variance using the formula:$$\text{Var}(X)=({100}^2)(0.9975)+(-2900{)}^2(0.002)+(-9900{)}^2(0.0005)-E(X{)}^2$$Calculate each term:- ${100}^2(0.9975)=9975$- $-{2900}^2(0.002)=16820$- $-{9900}^2(0.0005)=49005$Sum these:$$\text{Var}(X)=9975+16820+49005-{89}^2=69456-7921=61535$$The standard deviation is:$$\sigma =\sqrt{61535}\approx 248$$The standard deviation is approximately $248$.

Key concepts

Expected Profit

When students hear the term 'expected profit,' they might wonder how it differs from regular profit. The expected profit is not an actual profit. It's a projection of average outcomes based on different possible results and their probabilities. In this scenario, the insurance company sells a policy that involves different possibilities: no injury, minor injury, and major injury.

To calculate expectation, multiply the profit from each situation by its probability and sum these up. This gives the average or 'expected' profit over many policies, even if each policy might have different actual outcomes. Here’s a breakdown of how we achieve the expected profit of \(89 per policy:

• No Injury: Probability is 0.9975, Profit is \)100. So, this contributes to $100\times 0.9975=99.75$ to the expected profit.
• Minor Injury: Probability is 0.002, Profit is -$2900.Soitadds$-2900 \, \times \, 0.002 = -5.8\(.
• Major Injury: Probability is 0.0005, Profit is -\)9900. This is $-9900\times 0.0005=-4.95$.

Putting it together: $$E(X)=99.75-5.8-4.95=89$$ This means, on average, the company makes $89 on each policy sold, assuming their probability estimates are accurate.

Standard Deviation

Standard deviation helps us understand the risk or variability in potential profits. It tells us how much the actual profit could differ from the expected profit. A higher standard deviation means more potential variability, highlighting more risk. This is crucial for companies when assessing financial strategies.

In our example, the standard deviation measures how much the profit on policies varies due to the unpredictable nature of accidents. We calculate it by first finding the variance. To find variance ($$\text{Var}(X)$$), you calculate the squared difference from the expected profit for every outcome, multiply by their probabilities, and then sum them.

• No Injury: Profit deviation $100-89$ squared ${100}^2(0.9975)$
• Minor Injury: Profit deviation $-2900-89$ squared $(-2900{)}^2(0.002)$
• Major Injury: Profit deviation $-9900-89$ squared $(-9900{)}^2(0.0005)$

Add these squares and subtract $E(X{)}^2$ to get the variance of 61535, then take the square root for standard deviation:$$\sigma =\sqrt{61535}\approx 248$$This standard deviation of approximately 248 indicates a moderate level of risk associated with the policy.

Variance

Variance provides a deeper insight into the spread of possible outcomes, effectively quantifying the uncertainty of the expected profit. Unlike standard deviation, which is in the same unit as the data, variance is in squared units which are not as intuitive but offer analytical benefits.

The formula for variance involves finding how much each outcome deviates from the expected profit, squaring that difference, multiplying by the probability of the outcome, and then summing up those values. This gives us a measure of dispersion. For our insurance policy scenario:

• No Injury: Profit is 100, deviation from expected profit is minimal.
• Minor Injury: Profit is heavily negative (-2900), hence a larger squared deviation contribution.
• Major Injury: Profit is even more negative (-9900), hence the largest squared deviation contribution.

Let's recap the calculations:

• No Injury: ${100}^2(0.9975)=9975$
• Minor Injury: $(-2900{)}^2(0.002)=16820$
• Major Injury: $(-9900{)}^2(0.0005)=49005$

Add these squared contributions and subtract the square of the expected profit: $$\text{Var}(X)=9975+16820+49005-{89}^2$$ Equals 61535. Variance, therefore, illustrates the influences that cause profit to differ and the extent of impact, aiding companies in better managing and planning their financial risks.