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QuestionA company estimates a 0.3%0.3\% product failure rate after warranty, costing \400.Ifa2−yearwarrantycosts$57,findtheprofitvalues400. If a 2-year warranty costs \$57, find the profit values x,theirprobabilities, their probabilities P(x)$, and the expected value per warranty sold, rounded to the nearest cent.

Studdy Solution

STEP 1

Assumptions1. The failure rate of the products after the original warranty period but within years of the purchase is0.3% . The replacement cost for a failed product is 4003.Thecompanyoffersa−yearextendedwarrantyfor4003. The company offers a-year extended warranty for 574. The profit (denoted as xx) could be negative in the case of a loss5. We are looking at this problem from the company's point of view

STEP 2

First, we need to identify the possible outcomes and their corresponding probabilities. In this case, there are two possible outcomes1. The product fails and the company has to replace it. This happens with a probability of0.%.
2. The product does not fail and the company does not have to replace it. This happens with a probability of100% -0.% =99.7%.

STEP 3

Next, we calculate the profit for each outcome.1. If the product fails, the company has to pay the replacement cost of 400,buttheyhavereceived400, but they have received 57 from selling the warranty. So, the profit is 57−57 - 400 = -343.<br/>2.Iftheproductdoesnotfail,thecompanydoesnothavetopayanything,andtheyhavereceived343.<br />2. If the product does not fail, the company does not have to pay anything, and they have received 57 from selling the warranty. So, the profit is $57.

STEP 4

Now, we can create the probability distribution table.\begin{array}{|c|c|} \hlinex &(x) \\ \hline-\$343 &0.003 \\ \$57 &0.997 \\ \hline\end{array}

STEP 5

The expected value of a random variable is calculated as the sum of the product of each outcome and its corresponding probability. In this case, the expected value of the profit, (x)(x), is calculated as follows(x)=x1⋅(x1)+x2⋅(x2)(x) = x1 \cdot(x1) + x2 \cdot(x2)

STEP 6

Plug in the values for the profit and their corresponding probabilities to calculate the expected value.
(x)=−$343⋅0.003+$57⋅0.997(x) = -\$343 \cdot0.003 + \$57 \cdot0.997

STEP 7

Calculate the expected value of the profit.
(x)=−$343⋅0.003+$57⋅0.997=$56.27(x) = -\$343 \cdot0.003 + \$57 \cdot0.997 = \$56.27The company's expected value of each warranty sold is $56.27.

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