Solved on Jan 19, 2024

A company estimates $0.9\%$ of products fail within 2 years after warranty. Replacement cost is $\$ 500$ . What is the expected value of a $\$ 80$ 2-year extended warranty?

STEP 1

Assumptions
1. The failure rate of the products after the original warranty period but within 2 years is $0.9\%$ .
2. The replacement cost for a failed product is $\$500$ .
3. The cost of the extended warranty is $\$80$ .
4. The expected value is calculated as the average outcome, considering the probabilities of the different possible outcomes.

STEP 2

We need to calculate the expected cost to the company for each product that fails after the original warranty period but within 2 years. This is done by multiplying the probability of failure by the replacement cost.
$Expected\, cost\, of\, failure = Failure\, rate \times Replacement\, cost$

STEP 3

Convert the failure rate percentage to a decimal value.
$0.9\% = \frac{0.9}{100} = 0.009$

STEP 4

Now, plug in the given values for the failure rate and the replacement cost to calculate the expected cost of failure.
$Expected\, cost\, of\, failure = 0.009 \times \$500$

STEP 5

Calculate the expected cost of failure.
$Expected\, cost\, of\, failure = 0.009 \times \$500 = \$4.50$

STEP 6

Next, we need to calculate the expected profit from selling one extended warranty. This is the cost of the warranty minus the expected cost of failure.
$Expected\, profit = Warranty\, cost - Expected\, cost\, of\, failure$

STEP 7

Plug in the values for the warranty cost and the expected cost of failure to calculate the expected profit.
$Expected\, profit = \$80 - \$4.50$

STEP 8

Calculate the expected profit.
$Expected\, profit = \$80 - \$4.50 = \$75.50$
The company's expected value of each warranty sold is $\$75.50$ .

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A company estimates $0.9\%$ of products fail within 2 years after warranty. Replacement cost is $\$ 500$ . What is the expected value of a $\$ 80$ 2-year extended warranty?

STEP 1

STEP 2

We need to calculate the expected cost to the company for each product that fails after the original warranty period but within 2 years. This is done by multiplying the probability of failure by the replacement cost.
$Expected\, cost\, of\, failure = Failure\, rate \times Replacement\, cost$

STEP 3

Convert the failure rate percentage to a decimal value.
$0.9\% = \frac{0.9}{100} = 0.009$

STEP 4

Now, plug in the given values for the failure rate and the replacement cost to calculate the expected cost of failure.
$Expected\, cost\, of\, failure = 0.009 \times \$500$

STEP 5

Calculate the expected cost of failure.
$Expected\, cost\, of\, failure = 0.009 \times \$500 = \$4.50$

STEP 6

Next, we need to calculate the expected profit from selling one extended warranty. This is the cost of the warranty minus the expected cost of failure.
$Expected\, profit = Warranty\, cost - Expected\, cost\, of\, failure$

STEP 7

Plug in the values for the warranty cost and the expected cost of failure to calculate the expected profit.
$Expected\, profit = \$80 - \$4.50$

STEP 8

Calculate the expected profit.
$Expected\, profit = \$80 - \$4.50 = \$75.50$
The company's expected value of each warranty sold is $\$75.50$ .

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A company estimates 0.9%0.9\%0.9% of products fail within 2 years after warranty. Replacement cost is $500\$ 500$500. What is the expected value of a $80\$ 80$80 2-year extended warranty?

STEP 1

STEP 2

STEP 3

Convert the failure rate percentage to a decimal value.0.9%=0.9100=0.0090.9\% = \frac{0.9}{100} = 0.0090.9%=1000.9​=0.009

STEP 4

Now, plug in the given values for the failure rate and the replacement cost to calculate the expected cost of failure.Expected cost of failure=0.009×$500Expected\, cost\, of\, failure = 0.009 \times \$500Expectedcostoffailure=0.009×$500

STEP 5

Calculate the expected cost of failure.Expected cost of failure=0.009×$500=$4.50Expected\, cost\, of\, failure = 0.009 \times \$500 = \$4.50Expectedcostoffailure=0.009×$500=$4.50

STEP 6

STEP 7

Plug in the values for the warranty cost and the expected cost of failure to calculate the expected profit.Expected profit=$80−$4.50Expected\, profit = \$80 - \$4.50Expectedprofit=$80−$4.50

STEP 8

Calculate the expected profit.Expected profit=$80−$4.50=$75.50Expected\, profit = \$80 - \$4.50 = \$75.50Expectedprofit=$80−$4.50=$75.50The company's expected value of each warranty sold is $75.50 \$75.50$75.50.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

A company estimates $0.9\%$ of products fail within 2 years after warranty. Replacement cost is $\$ 500$ . What is the expected value of a $\$ 80$ 2-year extended warranty?

Convert the failure rate percentage to a decimal value.
$0.9\% = \frac{0.9}{100} = 0.009$

Now, plug in the given values for the failure rate and the replacement cost to calculate the expected cost of failure.
$Expected\, cost\, of\, failure = 0.009 \times \$500$

Calculate the expected cost of failure.
$Expected\, cost\, of\, failure = 0.009 \times \$500 = \$4.50$

Plug in the values for the warranty cost and the expected cost of failure to calculate the expected profit.
$Expected\, profit = \$80 - \$4.50$

Calculate the expected profit.
$Expected\, profit = \$80 - \$4.50 = \$75.50$
The company's expected value of each warranty sold is $\$75.50$ .