A company estimates that 0.7% of their products will fail after the original warranty period but within 2 years of the purchase, with a replacement cost of $350.  If they offer a 2 year extended warranty for $48, what is the company's expected value of each warranty sold?

To calculate the company's expected value of each extended warranty sold, we need to consider both the probability of a product failing within the extended warranty period and the replacement cost, as well as the revenue from selling the warranty. Here's a step-by-step breakdown:

1. **Given Data**:
   - Probability that a product will fail within 2 years after the original warranty period: $ p = 0.007 $ (or 0.7%)
   - Cost to replace a product if it fails: $350
   - Revenue from selling the extended warranty: $48

2. **Expected Loss Due to Replacement**:
   The expected cost for the company if a product fails is given by:
   \[
   \text{Expected cost} = p \times \text{Replacement cost}
   \]
   Substituting the values:
   \[
   \text{Expected cost} = 0.007 \times 350 = 2.45
   \]

3. **Expected Value Calculation**:
   The expected value of each warranty sold is the revenue from the warranty minus the expected cost of replacement:
   \[
   \text{Expected value} = \text{Revenue} - \text{Expected cost}
   \]
   Substituting the values:
   \[
   \text{Expected value} = 48 - 2.45 = 45.55
   \]

Thus, the company's expected value for each warranty sold is **$45.55**.

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### Related Questions:
1. How would the expected value change if the failure probability increased?
2. What if the replacement cost was $500 instead of $350?
3. How would the company's overall profit be affected if they sold 1,000 warranties?
4. What would be the break-even point if the warranty price was reduced?
5. How could the company adjust its pricing strategy if more customers purchase the warranty?

### Tip:
When calculating expected values in real-life situations, always consider both the costs and the probabilities involved to accurately assess the potential gains or losses.

A company estimates that 0.7% of their products will fail after the original warranty period but within 2 years of the purchase, with a replacement cost of $350. If they offer a 2 year extended warranty for $48, what is the company's expected value of each warranty sold?

Discover how to calculate the expected value of an extended warranty given a 0.7% failure rate and a $350 replacement cost. Learn about probability and expected value concepts, suitable for Grades 10-12.

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