A company estimates that \( 7 \% \) of their products will fail after the original warranty period but within 2 years of the purchase, with a replacement cost of \( \$ 250 \). If they want to offer a 2 year extended warranty, what price should they charge so that they'll break even (in other words, so the expected value will be 0 )

To break even, the company needs to calculate the expected costs of replacements covered by the extended warranty. If \( 7\% \) of products are expected to fail, the expected cost per product is: \[ \text{Expected Cost} = 0.07 \times 250 = 17.5 \] Therefore, to cover this expected cost through the price of the extended warranty, the company should charge \( \$17.50 \) for the 2-year extended warranty. This ensures that the expected payout and the income from the warranty fees balance each other out, leading to an expected value of 0. If they want to add a bit of profit, they can simply adjust the price upwards!