Problem 29 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

INSURANCE POLICY An insurance company charges \(\$ 10,000\) for a policy insuring against a certain kind of accident and pays \(\$ 100,000\) if the accident occurs. Suppose it is estimated that the probability of the accident occurring is \(p=0.02\). Let \(X\) be the random variable that measures the insurance company's profit on each policy it sells. a. What is the probability distribution for \(X\) ? b. What is the company's expected profit per policy sold? c. What should the company charge per policy to double its expected profit per policy?

Step-by-Step Solution

Answer

a. \( P(X = 10,000) = 0.98 \, P(X = -90,000) = 0.02 \). b. Expected profit is \$8,000\. c. Charge \$18,163.27\ per policy.

Step 1 Title - Define the Random Variable

Define the random variable \(X\) representing the insurance company's profit on each policy. The profit depends on whether an accident occurs or not. There are two scenarios - if no accident occurs (\textbackslash\( X = 10,000 \textbackslash\)) and if an accident occurs (\textbackslash\( X = 10,000 - 100,000 = -90,000 \textbackslash\)).

Step 2 Title - Determine Probability Distribution

a. Determine the probability distribution for \(X\): \[ P(X = 10,000) = 1 - p = 0.98 \] \[ P(X = -90,000) = p = 0.02 \]

Step 3 Title - Expected Profit Calculation

b. Calculate the company's expected profit per policy, \( E(X) \): \[ E(X) = 10,000 \times P(X = 10,000) + (-90,000) \times P(X = -90,000) \] \[ E(X) = 10,000 \times 0.98 + (-90,000) \times 0.02 \] \[ E(X) = 9,800 + (-1,800) \] \[ E(X) = 8,000 \]

Step 4 Title - New Charge to Double Expected Profit

c. Find the charge \(C\) to double the expected profit. The new expected profit should be \(16,000 \), so: \[ E(X_{new}) = 100,000C \times P(X = new profit) + (-90,000) \times P(X = -90,000) \] \[ 16,000 = C \times 0.98 + (-90,000) \times 0.02 \] \[ 16,000 = 0.98C - 1,800 \] \[ 16,000 + 1,800 = 0.98C \] \[ 17,800 = 0.98C \] \[ C = \frac{17,800}{0.98} \] \[ C = 18,163.27 \]

Key Concepts

probability distribution

In the context of insurance policy profit calculation, understanding the probability distribution is crucial for predicting outcomes. Here, we consider the profit the insurance company earns from each policy sold. Let's define two possible scenarios:

• If an accident does occur, the company loses money, resulting in a profit of \( -90,000 \) dollars per policy.
• If no accident occurs, the company makes a profit of \( 10,000 \) dollars per policy.

The probabilities of these outcomes must be calculated to form the probability distribution for the random variable \( X \). Given that the probability of an accident happening is \( p = 0.02 \), the probability of no accident happening is \( 1 - p = 0.98 \). Hence:
\[ P(X = 10,000) = 0.98 \]
\[ P(X = -90,000) = 0.02 \]
Probability distributions help determine the likelihood of different outcomes, enabling the company to make informed decisions.

expected profit

Expected profit provides an average outcome of a probabilistic event over numerous trials. In simple terms, it is the weighted average of all possible profits, where the weights are the probabilities of each outcome.
For the insurance policy, the expected profit \ E(X) \ is calculated as follows:
\[ E(X) = \text{(profit if no accident)} \times P(\text{no accident}) + \text{(loss if accident)} \times P(\text{accident})\]Plugging in the given numbers:
\[ E(X) = 10,000 \times 0.98 + (-90,000) \times 0.02\]
Calculating this, we get:
\[ E(X) = 9,800 + (-1,800) = 8,000 \]
The expected profit per policy sold is \( 8,000 \) dollars. This metric allows the insurance company to predict its average profitability from selling these policies.

charge per policy

Calculating the correct charge per policy is essential to ensure profitability aligns with business goals. If the company wants to double its expected profit per policy, we need to find the new charge that achieves this. Let's denote the new charge as \( C \). We want to make the expected profit per policy equal to \( 16,000 \).
The equation will be set up as:
\[ E(X_{new}) = \text{(new charge)} \times P(\text{no accident}) + (\text{loss if accident}) \times P(\text{accident})\]
Substitute in the known values:
\[ 16,000 = C \times 0.98 + (-90,000) \times 0.02\]
We solve for \( C \) by isolating it in the equation:
\[ 16,000 = 0.98C - 1,800 \text \16,000 + 1,800 = 0.98C \17,800 = 0.98C \C = \frac{17,800}{0.98} \C = 18,163.27 \]
Thus, the company should charge approximately \( 18,163.27 \) dollars per policy to double its expected profit.

random variable

A random variable is a numerical representation of the outcomes of a random phenomenon. In the context of the insurance policy, our random variable \( X \) represents the profit the company makes from selling one policy.
Here, \ X \ can take two possible values:

• x = 10,000
• x = -90,000

This is because the company either makes a profit if no accident occurs ( \ X = 10,000 \ ), or it makes a loss if an accident occurs ( \ X = -90,000 \ ).
Considering the probabilities:

• P(X = 10,000) = 0.98
• P(X = -90,000) = 0.02

Therefore, the random variable model helps in summarizing the possible outcomes of selling a single policy, providing a probabilistic framework for analysis.