A company estimates that 0.5% of their products will fail after the original warranty period but within 2 years of the purchase, with a replacement cost of $200. If they offer a 2 year extended warranty for $11, what is the company's expected value of each warranty sold?

The company is expected to make on every extended warranty sold.

Step-by-step explanation:

Let denote whether a product with the extended warranty requires replacement in that two years. ( would be a random variable.) Assume that means that the product requires replacement, and otherwise.

Assume that this product requires replacement in that two years. That is: . The company would then bear a cost of for replacing this product (since , that cost would be the same as .)

On the other hand, assume (that is: this product does not require replacement in that two years.) The company would not need to pay for replacing this product. Since , the expression would still represent the cost for the company for this warranty.  

Either way, would denote the replacement cost that the company would bear for this product. However, given that the company would charge for the extended warranty, the net revenue of the company on this warranty would be . (An earning of minus a spending of .)

The question states that of the products would need replacement in this period. In other words, the expected value of would be .

The expected revenue of the company on this warranty would be:

.

Apply the linearity of expected values to find the value of :

.

Hence, the company is expected to make on every extended warranty that it sold.

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