https://file.aiquickdraw.com/mathbot/file/1730053023590sw0r2kw0.jpg

To solve this, we need to determine the expected value of the cost MotoWin will incur for each warranty sold, given the probabilities and costs involved.

Here’s the information we have:

1. **Cost of the warranty**: $993
2. **Cost of replacing the car**: $12,200
3. **Probability of needing a replacement**: 8% (or 0.08)
4. **Probability of not needing a replacement**: 92% (or 0.92)

### Expected Cost Calculation
The expected cost for MotoWin per warranty is calculated by considering both scenarios (replacement needed and no replacement needed), weighted by their probabilities.

\[
\text{Expected Cost} = (0.08 \times 12,200) + (0.92 \times 0)
\]
\[
= (0.08 \times 12,200) + 0
\]
\[
= 976
\]

### Expected Profit or Loss
Now, we compare the revenue from the warranty ($993) to the expected cost ($976).

\[
\text{Expected Profit} = 993 - 976 = 17
\]

Since the expected profit is positive, MotoWin can expect to make money on each warranty sold. Therefore, the correct answer is:

**"MotoWin can expect to make money from offering these warranties. In the long run, they should expect to make $17 on each warranty sold."**

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Would you like further details or have any questions?

### Additional Questions for Deeper Understanding:
1. What if the replacement cost changed to a different value?
2. How would the expected profit change if the probability of needing replacement increased to 10%?
3. If MotoWin offered a discount on the warranty, how much could it go down without losing money?
4. What impact would an increase in car reliability (lower probability of needing replacement) have on profits?
5. How would the calculations change if multiple replacements were covered under the warranty?

### Tip
In probability-based problems, expected value helps you evaluate long-term outcomes by considering each possible scenario's cost or gain multiplied by its probability.

If MotoWin knows that it will sell many of these warranties, should it expect to make or lose money from offering them? How much?

Calculate the expected profit or loss for MotoWin Auto's two-year limited warranty offer at $993, considering a replacement cost of $12,200 and an 8% chance of replacement. Suitable for high school students learning expected value and cost analysis.

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