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Chapter 4: Problem 27

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 \(A ?\)

Short Answer

Expert verified
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 by step solution

01

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.
02

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)
03

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)
04

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.

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