A company estimates that 
 of their products will fail after the original warranty period but within two years of the purchase, with a replacement cost of 
.

If they offer a two-year extended warranty for 
, what is the company's expected value of each warranty sold?

Complete the table with the appropriate values.  Be sure to include negatives where appropriate.

Outcomes	Product fails within two years	Product does not fail within two years
Probability of each outcome	
Payoff of each outcome ($)	
Use the table to calculate the expected value.  Express your answer to the nearest cent.

Company's expected value of the two-year extended warranty = $
Incorrect  

Hint: Be sure to convert your percents to decimal form, i.e. `0.3% = 0.3/100 = 0.003.

Let's walk through the problem systematically to calculate the expected value (EV) of the extended warranty.

### Step-by-Step Breakdown:

1. **Define the Variables:**
   - Let $ p_f $ be the probability that a product fails within two years.
   - Let $ C_r $ be the replacement cost if the product fails.
   - Let $ W $ be the cost of the two-year extended warranty.

The information given is as follows (you might need to fill in the missing values based on the data):
- $ p_f $ is the probability of product failure within two years.
- The replacement cost $ C_r $ is the cost to the company if the product fails.
- The extended warranty cost $ W $ is the amount the company charges customers for the extended warranty.

2. **Table Setup:**
   We need to complete a table that captures the following:

| Outcome                             | Product fails within two years | Product does not fail within two years |
|-------------------------------------|-------------------------------|----------------------------------------|
| **Probability of each outcome**     | $ p_f $                      | $ 1 - p_f $                         |
| **Payoff of each outcome ($)**      | $ -C_r + W $                 | $ W $                               |

Where:
- $ p_f $ is the probability of failure.
- $ C_r $ is the cost of replacing a failed product (which is a loss for the company).
- $ W $ is the revenue the company receives from selling the extended warranty.

3. **Expected Value (EV) Formula:**
The expected value for the company is calculated as:
\[
\text{EV} = (p_f \times (\text{Payoff when the product fails})) + ((1 - p_f) \times (\text{Payoff when the product does not fail}))
\]
Substituting the payoffs from the table:
\[
\text{EV} = p_f \times (-C_r + W) + (1 - p_f) \times W
\]

4. **Fill in Missing Data and Calculate:**
You need to insert the actual numbers for $ p_f $, $ C_r $, and $ W $. Once those are provided, you can plug them into the formula and calculate the expected value of the two-year extended warranty.

Would you like to provide the missing values for $ p_f $, $ C_r $, and $ W $ so I can complete the calculation?

A company estimates that a percentage of their products will fail after the original warranty period but within two years of the purchase, with a replacement cost of $X. If they offer a two-year extended warranty for $Y, what is the company's expected value of each warranty sold?
Complete the table with the appropriate values. Be sure to include negatives where appropriate.

| Outcomes                            | Product fails within two years | Product does not fail within two years |
|-------------------------------------|-------------------------------|----------------------------------------|
| **Probability of each outcome**     |                               |                                        |
| **Payoff of each outcome ($)**      |                               |                                        |

Use the table to calculate the expected value. Express your answer to the nearest cent.

Company's expected value of the two-year extended warranty = $

Learn how to calculate the expected value of a two-year extended warranty for a company, given probabilities of product failure and associated costs. This problem involves key concepts in probability and decision theory, suitable for high school students in Grades 9-12.

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