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The problem states:

- A company estimates that **0.2%** (or 0.002 in decimal form) of their products will fail after the original warranty period but within 2 years of purchase.
- The replacement cost is **$350** per failed product.
- The company offers a **2-year extended warranty for $22**.

The question asks: What is the company's expected value of each warranty sold?

### Solution:
The expected value (EV) of each warranty is based on two possible outcomes:

1. **The product does not fail (with probability 99.8%)**:
   - In this case, the company earns $22 for the warranty but does not have to pay anything.
   - The value to the company is $22.

2. **The product fails (with probability 0.2%)**:
   - The company earns $22 for the warranty but has to pay $350 for the replacement.
   - The value to the company is $22 - $350 = **-$328** (a loss).

#### Calculation of Expected Value (EV):
\[
EV = (0.998 \times 22) + (0.002 \times -328)
\]

Breaking this down:

- $ 0.998 \times 22 = 21.956 $
- $ 0.002 \times -328 = -0.656 $

\[
EV = 21.956 - 0.656 = 21.3
\]

Thus, the company's **expected value for each warranty sold** is **$21.30**.

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Do you need more details or have any questions?

Here are some additional related questions:
1. How would the expected value change if the failure rate was higher?
2. What would happen to the expected value if the warranty price increased?
3. How would a lower replacement cost affect the expected value calculation?
4. What is the probability threshold where the company starts losing money on warranties?
5. How could the company mitigate the risks associated with product failures?

**Tip**: The expected value helps businesses predict long-term outcomes and profitability based on statistical probabilities!

A company estimates that 0.2% 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 $22, what is the company's expected value of each warranty sold?

Calculate the expected value of each extended warranty sold, given a 0.2% failure rate, a $350 replacement cost, and a $22 warranty price. Understand the use of probability in determining profit expectations in business.

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