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QuestionA company expects 0.7%0.7\% of products to fail after warranty, costing \250.Theysella$39warranty.Findprofit250. They sell a \$39 warranty. Find profit x$ values and expected value.

Studdy Solution

STEP 1

Assumptions1. The failure rate of the products after the original warranty period but within years is0.7% . The replacement cost for each failed product is 2503.Thecompanyoffersa−yearextendedwarrantyfor2503. The company offers a-year extended warranty for 394. The profit xx could be negative in the case of a loss5. We need to complete the probability distribution table and find the company's expected value for each warranty sold

STEP 2

First, we need to identify the possible outcomes and their probabilities. There are two outcomes the product fails and the company has to replace it (0.7% chance), or the product doesn't fail and the company doesn't have to replace it (99.% chance).
(Failure)=0.7%=0.007(Failure) =0.7\% =0.007(No Failure)=100%−0.7%=99.%=0.993(No\, Failure) =100\% -0.7\% =99.\% =0.993

STEP 3

Next, we calculate the profit in each case. If the product fails, the company loses the replacement cost but keeps the warranty price. If the product doesn't fail, the company keeps the warranty price.
Profit(Failure)=Warranty price−Replacement cost=$39−$250=−$211Profit(Failure) = Warranty\, price - Replacement\, cost = \$39 - \$250 = -\$211Profit(No Failure)=Warranty price=$39Profit(No\, Failure) = Warranty\, price = \$39

STEP 4

Now, we can create the probability distribution table\begin{array}{|c|c|} \hlinex &(x) \\ \hline-\$211 &0.007 \\ \$39 &0.993 \\ \hline\end{array}

STEP 5

Finally, we calculate the expected value of the profit. The expected value is the sum of each outcome multiplied by its probability.
(x)=∑x⋅(x)(x) = \sum x \cdot(x)

STEP 6

Plug in the values for the profit and their probabilities to calculate the expected value.
(x)=−$211⋅0.007+$39⋅0.993(x) = -\$211 \cdot0.007 + \$39 \cdot0.993

STEP 7

Calculate the expected value of the profit.
(x)=−$211⋅0.007+$39⋅0.993=$38.26(x) = -\$211 \cdot0.007 + \$39 \cdot0.993 = \$38.26The company's expected value for each warranty sold is $38.26.

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