Problem 27 - A First Course in Probability

Question

An insurance company writes a policy to the effect that an amount of money \(A\) must be paid if some event \(E\) occurs within a year. If the company estimates that \(E\) will occur within a year with probability \(p,\) what should it charge the customer in order that its expected profit will be 10 percent of \(A ?\)

Step-by-Step Solution

Answer

The insurance company should charge the customer \(\frac{0.1A}{1 - p^2}\) in order that its expected profit will be 10% of A.

Step 1: Represent the profit

First, let's represent the profit the insurance company makes in the two possible scenarios: E occurs or E does not occur. Let x represent the amount charged to the customer. If E does not occur, the company keeps the whole amount charged, which results in a profit of x. If E occurs, the company has to pay out A, and the profit will be x - A.

Step 2: Calculate expected profit

The expected profit is calculated by multiplying the profit for each scenario by its respective probability and then adding them up. Expected profit = (probability of E not occurring) * (profit if E does not occur) + (probability of E occurring) * (profit if E occurs) Let's denote the probability of E not occurring as q. Then q = 1 - p, since the probabilities of all possible outcomes must add up to 1. Expected profit = q(x) + p(x - A)

Step 3: Set expected profit equal to 10% of A and solve for x

The problem states that the expected profit should be equal to 10% of A, or 0.1A. Plugging this into our formula for expected profit: 0.1A = q(x) + p(x - A) Now we can plug in q = 1 - p and solve for x: 0.1A = (1 - p)(x) + p(x - A)

Step 4: Simplify and solve for x

Let's simplify the equation and solve for x: 0.1A = x - xp + px - p^2x 0.1A = x - p^2x Now, factor x out: 0.1A = x(1 - p^2) Now, divide by (1 - p^2) to get the result for x: x = \(\frac{0.1A}{1 - p^2}\) The insurance company should charge the customer \(\frac{0.1A}{1 - p^2}\) in order that its expected profit will be 10% of A.